Embed Size (px)
DESCRIPTION
standarde mixturi
Citation preview
EUROPEAN STANDARD
NORME EUROPÉENNE
EUROPÄISCHE NORM
DRAFTprEN 13036-5
January 2006
ICS
English Version
Road and airfield surface characteristics - Test methods - Part 5:Determination of longitudinal unevenness indices
Oberflächeneigenschaften von Straßen und Flugplätzen -Prüfverfahren - Teil 5: Bestimmung der
Längsunebenheitsindizes
This draft European Standard is submitted to CEN members for enquiry. It has been drawn up by the Technical Committee CEN/TC 227.
If this draft becomes a European Standard, CEN members are bound to comply with the CEN/CENELEC Internal Regulations whichstipulate the conditions for giving this European Standard the status of a national standard without any alteration.
This draft European Standard was established by CEN in three official versions (English, French, German). A version in any other languagemade by translation under the responsibility of a CEN member into its own language and notified to the Management Centre has the samestatus as the official versions.
CEN members are the national standards bodies of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France,Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania,Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
Recipients of this draft are invited to submit, with their comments, notification of any relevant patent rights of which they are aware and toprovide supporting documentation.
Warning : This document is not a European Standard. It is distributed for review and comments. It is subject to change without notice andshall not be referred to as a European Standard.
EUROPEAN COMMITTEE FOR STANDARDIZATIONC OM ITÉ EUR OP ÉEN DE NOR M ALIS AT IONEUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36 B-1050 Brussels
© 2006 CEN All rights of exploitation in any form and by any means reservedworldwide for CEN national Members.
Ref. No. prEN 13036-5:2006: E
prEN 13036-5:2006 (E)
2
Contents Page
Introduction .........................................................................................................................................................3 1 Scope ......................................................................................................................................................3 2 Normative references ............................................................................................................................4 3 Terms and definitions ...........................................................................................................................4 4 Symbols and abbreviations ..................................................................................................................6 5 Computation of unevenness indices...................................................................................................7 5.1 General overview of the computation .................................................................................................7 5.2 Pre-filtering and re-sampling................................................................................................................8 6 Computation of unevenness indices.................................................................................................11 6.1 International Roughness Index (IRI) ..................................................................................................11 6.1.1 General..................................................................................................................................................11 6.1.2 Moving average length........................................................................................................................12 6.1.3 Presentation base length ....................................................................................................................12 6.2 Wave bands analysis...........................................................................................................................12 6.2.1 Computation of the indices ................................................................................................................12 6.2.2 Filters ....................................................................................................................................................13 6.2.3 Filtering algorithm ...............................................................................................................................14 6.2.4 Wave band indices...............................................................................................................................15 6.3 PSD analyses .......................................................................................................................................16 6.3.1 Segmentation and windowing............................................................................................................16 6.3.2 PSD Computation ................................................................................................................................16 6.3.3 De-colouring.........................................................................................................................................16 6.3.4 Smoothing ............................................................................................................................................16 6.3.5 Fitting and computation......................................................................................................................16 7 Test report ............................................................................................................................................18 Annex A (informative) Re-sampling.................................................................................................................19 Annex B (informative) Wave bands analysis..................................................................................................20 B.1 Re-sampling and three bands filtering ..............................................................................................20 B.1.1 Re-sampling and three bands filtering process detailed algorithm...............................................22 B.2 Detailed characteristics of filters .......................................................................................................24 Annex C (informative) An illustration of PSD computation ..........................................................................26 C.1 General..................................................................................................................................................26 C.2 Segmentation and windowing............................................................................................................26 C.3 PSD Computation ................................................................................................................................27 C.4 De-colouring.........................................................................................................................................27 C.5 Smoothing ............................................................................................................................................27 C.6 Fitting and computation......................................................................................................................30 Bibliography ......................................................................................................................................................35
prEN 13036-5:2006 (E)
3
Foreword
This document (prEN 13036-5:2006) has been prepared by Technical Committee CEN/TC 227 “Road materials”, the secretariat of which is held by DIN.
This document is currently submitted to the CEN Enquiry.
Introduction
The road profile unevenness through road/vehicle dynamic interaction and vehicle vibration affects safety (tyre contact forces), ride comfort, energy consumption and vehicle wear. The road profile unevenness is consequently a key information for road maintenance management systems.
The measurement of road unevenness has been a subject of numerous researches for more than 70 years. Methods developed can be classified into two types:
those based on response type devices and
those based on profiling devices or profilometers.
Assessing the condition of a road using a profilometer usually involve to record its profile, then computing a limited set of numbers or indices characterising the unevenness, and eventually comparing these indexes to a reference scale. Only profilometers able to digitise and record under a digital format a road profile from which different indices can be computed are considered in this prEN 13036-6.
The purpose of this document is to standardise various possible characterisations of the road profile unevenness such as the International Roughness Index (IRI) computation procedure, wave bands analyses as well as Power Spectral Density (PSD) analyses. The objective of the document is not to impose a single specific procedure but to insure that when applying one of the possible procedure exactly the same steps are carried out with the aim of facilitating the comparison of unevenness measurements carried out with different profiling instruments in European countries.
It is beyond the scope of this document to provide reference values for these indices, or to provide detailed information about the characteristics of profilometers.
1 Scope
This document defines different possible methods for processing digitised road profiles:
Computation of the International Roughness Index (IRI); based on the Golden car characteristics,
Spectral analyses: Wave band analysis and Spectrum analysis, based on the Power Spectral Density (PSD)
prEN 13036-5:2006 (E)
4
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.
prEN 13036-6, Road and airfield surface characteristics — Test methods — Part 6: Measurement of transverse and longitudinal profiles in the unevenness and megatexture wavelength ranges.
ISO 2041, Vibration and shock — Vocabulary.
ISO 8608, Mechanical vibrations — Road surface profiles — Reporting of measured data.
ANSI – S1. 11-2004-07-27, Specification for octave band and fractional octave band, analog and digital filters.
IEC 61260, Octave-band and fractional-octave-band filters.
ISO TS 13473-4, Characterisation of pavement texture by use of surface profiles. (in preparation)
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
3.1 profile is the intersection between the surface of the pavement and the plane which contains both the vertical of the measured pavement and the line of travel of the measuring instrument; when the measuring instrument travels in a curve the line of travel is the tangent to that curve, when travelling in a straight line it is this line. In this plane, a point of the profile can be adequately described by its coordinates x (abscissa) and z (elevation), in any orthonormal reference system (X, Z), where Z is parallel to the aforementioned vertical
3.2 spatial sampling interval is the absolute value of the difference of abscissa between two adjacent points of the digitised longitudinal profile line. This definition assumes that the distance measured by the profilometer, which is usually related to the curvilinear abscissa, is close enough to the abscissa in the mathematical sense
3.3 longitudinal road profile is one of the profiles obtained when the measuring instrument travels in the same direction as the usual traffic. Usually one of the profiles measured in the wheel tracks is used
NOTE Strictly speaking the digitised profile given by a profilometer, is a distorted image of the real profile, usually referred as a pseudo-profile ; in order to make the remaining part of this standard more easy to read the word profile is used to denote this image.
3.4 Longitudinal unevenness is the deviation of the longitudinal profile from a straight reference line in a wavelength range of 0,5 m to 50 m. The reference line, is usually the intersection of the profile plane and the horizontal plane.
The range from 0,5 m to 50 m is the common range for roads. This limit can be extended to 100 m for runways. Higher values don’t deal with unevenness but depend on road geometry
prEN 13036-5:2006 (E)
5
3.5 raw profile is the profile given by a profilometer when measuring a longitudinal road profile. Characteristics of raw profile, depend on the profilometer used
3.6 pre-processed profile (re-sampled and filtered profile) is obtained by applying the re-sampling and filtering procedure described in clause 6. The pre-processing procedure aims to harmonize the profile provided by various devices
3.7 wavelengths in most cases the profile can be adequately described as a sum of sine functions, when this is possible one such sine function is
( )
−
Λ 02 xxA πsin
Where
Λ is the wavelength of the sine in metres (m);
A is the amplitude of the sine in metres (m);
x is the abscissa of the current point, in metres (m);
x0 is the phase of the sine, in metres (m).
3.8 spatial frequency is the reciprocal of a wavelength in cycles per metre. The spatial frequency N defines the number of waves, of wavelength Λ, per metre:
Λ= 1N
3.9 spatial sampling interval is the horizontal distance between two adjacent points of the digitised longitudinal profile line
3.10 standard reference sampling interval is the spatial sampling interval which must be used when computing the indices defined in this standard, its value is 0,05 m
3.11 measuring track is the intersection of the envelope of the profile plane and the horizontal plane
3.12 profile measurement length is the length of an uninterrupted profile measurement. It is the length over which the profilometer continuously and accurately digitises and records the profile (from point B to C in Figure 1). Most profilometers need to run for some minimum distance before and after the very profile they are to measure, these starting (from point A to B in Figure 1) and ending phases (from point C to D in Figure 1) should not be included in the profile measurement length
prEN 13036-5:2006 (E)
6
Key 1 road 2 l0 . . . ln sample reporting length
3 profile measurement length 4 overall profilometre route
Figure 1 — Profile lengths definitions
3.13 evaluation or reporting length are the measurements made over the profile measurement length which are often analysed using shorter parts or samples (l0 to ln in Figure 1) to allow for a more precise description of the measured profile. The evaluation or reporting length is the length of such a sample
NOTE In the case of consecutive samples such as l0 and l1 in Figure 1, over the profile measurement length, the word “segment” is used.
3.14 PSD (Power Spectral Density) Is the limiting mean-square value of a signal spectrum per unit bandwidth, i.e. the limit of the mean-square value in a rectangular bandwidth divided by the bandwidth, as the bandwidth approaches zero. In the unevenness measurement field, the signal used is usually the road profile. In practice the PSD spectrum is characterised by fitted straight regression line[s] and expressed by indices related to the location of these line[s]
4 Symbols and abbreviations
Symbols, which are used in equation are written using normal characters, abbreviations are written using bold characters.
B is the base used for IRI computation in metre (m). It is the length over which the IRI computation is performed (or reporting length using the terminology of this document).
G(x) is the displacement PSD value for the spatial frequency x;
L denotes the measurement length, in metres (m), provided the conversion factor (1 km = 103 m) is given, kilometres can be used as an alternative;
N is the spatial frequency, in cycles per metre (m): Λ
= 1N ; N is usually called a wave number;
Xi is the abscissa of the sampled point i, in metre (m).
prEN 13036-5:2006 (E)
7
zi is the elevation of the profile determined at the sampling point i, in metre (m), provided the conversion factor (1 mm = 10–3 m) is given, millimetres can be used as an alternative;
δx is the spatial sampling interval for the digitisation of the profile, in metre (m), provided the conversion factor (1 mm = 10–3 m) is given, millimetres can be used as an alternative;
Λ is the wavelength, in metre (m);
Ω is the spatial frequency, in radian per metre (rad/m) Λ
=Ω π2 ;
G(Ω0) is the unevenness index; where Ω0 = 1 rad/m;
IRI is the International Roughness Index;
PSD is the Power Spectrum Density of a signal spectrum;
WB is the wave band index calculated by using root mean square analysis applied to the pre-processed profile elevations for the wave band W, in metre (m), provided the conversion factor (1 mm = 10–3 m) is given, millimetres can be used as an alternative;
SW is the Root Mean Square value of the short waves band;
MW is the Root Mean Square value of the medium waves band;
LW is the Root Mean Square value of the long waves band;
w is the waviness of the signal spectrum
5 Computation of unevenness indices
5.1 General overview of the computation
The computation of unevenness indices, involves three steps:
the measurement and pre-processing of the profile, the output of which is a filtered and re-sampled (or pre-processed) profile,
the computation of one or more index(es),
the creation of a report (see Figure 2).
prEN 13036-5:2006 (E)
8
Figure 2 — General overview of the computation
The pre-processing is essential in the case of wave band analysis is strongly recommended to homogenize the profiles and facilitate comparisons.
5.2 Pre-filtering and re-sampling
In order to allow for meaningful comparisons all the analyses described below, should be carried out using exactly the same algorithms which must be applied to signals sampled with exactly the same sampling interval.
As indicated in Figure 2, the output of a profilometer measurement is a raw profile, which makes use of a sampling interval which depends on the profilometer used.
NOTE The profilometer used should at least have a class 2 vertical definition and traveled distance accuracy, and a class 3 acquisition sampling interval, larger wavelength cutoff and reporting sampling interval as defined in the prEN 13036-6
As it is very unlikely that all profilometers will natively report results using the same sampling interval, one of the first step of the spectrum analysis must consist in re-sampling the original data to the standard reference sampling interval which is defined to be 0,05 m. This re-sampling process must be preceded by a bandpass filtering of the original signal, in order to insure that no unwanted distortion of the profile can be introduced (see Figure 3).
prEN 13036-5:2006 (E)
9
The result of this procedure is the pre-processed profile, which has a sampling interval of 0,05 m, and a wavelength bandwidth limited to the 0,781 m to 50,0 m band. For certain applications the limitation of bandwidth is not applied.
Unless the used profilometer and the computations made afterward make use of a spatial sampling interval which is an integer multiple of the standard reference sampling interval, carrying out this the pre-processing is mandatory.
Profile analysis can afterward be carried out using either limited wave bands and derived associated indices or the full frequency content of the signal in which case the spectral density must be adequately estimated and described.
prEN 13036-5:2006 (E)
10
Key 1 elevations, in millimetre (mm) 2 measured originals profile (x = sampled points) 3 filtered originals profile (x = sampled points) 4 resampled profile (x = original points, • = resampled points)
Figure 3 — Filtering and Re-Sampling illustration
prEN 13036-5:2006 (E)
11
6 Computation of unevenness indices
6.1 International Roughness Index (IRI)
6.1.1 General
The IRI is an index computed from a longitudinal road profile measurement using a virtual response type system, quarter-car simulation, running at a speed of 80 km/h, (see Figure 4). The quarter-car simulation applied on the digitised road profile calculates the accumulated suspension motions divided by the distance travelled. Using a continuous road profile this is illustrated in formula (1). Time is related to longitudinal distance by the simulated speed of the quarter-car simulation, t = x/V, where x is the longitudinal distance and V is the simulated forward speed.
dtzzB
T
∫ −=0
1IRI us && (1)
In practise all road profiles are sampled and the formula thus follows
∑=
−=n
iss
n 1
1IRI iu,is, (2)
The IRI has the unit of slope, e.g. mm/m or m/km. A complete computation of IRI is described in World Bank Technical Papers 45 and 46, [1], [2] and additional information in [3] and [4] as well as a complete computer code that is provided in [5].
Key 1 spring mass 2 suspension damping rate Cs
3 suspension spring rate Ks
4 unspring mass 5 tyre spring rate kt
6 longitudinal profile Z(x)
Figure 4 — Quarter car (virtual response type system)
prEN 13036-5:2006 (E)
12
6.1.2 Moving average length
IRI should be calculated using the state transition matrix principle, see reference [4]. A smoothed profile height determined with a moving average length of Ls = 250 mm should be used as input into the calculations. The sampling interval should be 125 mm or less. If the moving average is already applied when the road profile is measured it should not be done again. The averaging should only be done once.
By using the specific parameter values, Golden Carsee Table 1) and the speed of 80 km/h the output from the quarter-car simulation is defined as the International Roughness Index, IRI. The IRI scale starts at zero for a road with no unevenness and covers positive numbers that increases in proportion to unevenness.
Table 1 — The Golden car parameter values
K1 = kt / Ms = 653 s–2 k2 = ks / Ms = 63,3 s–2
u = mu / Ms = 0,15 c = Cs / Ms = 6,0 s–2
6.1.3 Presentation base length
The base length, B in formula (1), can be any length from the sample length and up; this is decided by the use of the IRI-value. For international comparison of IRI it is found, that a length of 100 meter is suitable [6]. The presentation base length must be reported together with the IRI value. It is important to remember that intervention thresholds depend on the presentation base length.
6.2 Wave bands analysis
6.2.1 Computation of the indices
In order to perform wave band analysis, the pre-processed profile is splitted into different wave band limited profile using filters (see Figure 5). The definition of the wave bands used as well as the characteristics of the filters used to obtain band limited signals, from the original longitudinal profile must be given. How indices are derived from the band limited signals must also be defined.
NOTE The chosen indices and associated computations are directly derived from the ones used in the FEHRL FILTER experiment at the European level, and the PIARC EVEN experiment at the world-wide level [7], [8], [9], [10].
prEN 13036-5:2006 (E)
13
Key 1 elevation, in millimetre (mm) 2 pre-processed profile 3 lowpass filter 4 bandpass filter
5 highpoass filter 6 long waves filtered profile 7 medium waves filtered profile 8 shortwaves filtered profile
Figure 5 — Wave bands splitting
6.2.2 Filters
6.2.2.1 Octave filters (according to ANSI-S1-11-2004-07-27)
Octave filters are constant relative band filters defined by
2=l
hff
(3)
where
fh is the high cutoff spatial frequency at – 3 dB, in cycles per metre;
fl is the low cutoff spatial frequency at – 3 dB, in cycles per metre.
The central spatial frequency of such a filter is defined as fc = hl ff .
prEN 13036-5:2006 (E)
14
6.2.2.2 Bi-octave band
The bi-octave bands used for the determination of band limited unevenness indices group together two consecutive octaves.
NOTE These bands have been defined for the FEHRL FILTER [9] and PIARC EVEN [10] projects.
These bands are defined below:
short wave SW band grouping together the octaves having a central spatial frequency of 1,280 cycle per metre and 0,320 cycle per metre;
medium wave MW band grouping together the octaves having a central spatial frequency of 0,320 cycle per metre and 0,080 cycle per metre;
long wave LW band grouping together the octaves having a central spatial frequency of 0,080 cycle per metre and 0,020 cycle per metre.
Table 2 summarises these values showing wavelengths as well as spatial frequencies.
Table 2 — Characteristics of the filters
Spatial filtering in wavelengths Band
Bi-octave m
Octave m
Spatial frequencies in cycles per metre
Short waves (SW)
low limit
central value
high limit
0,781
1,563
3,125
0,781 1,105 1,563 2,210 3,125
1,280 0,905 0,640 0,453 0,320
Medium waves (MW)
low limit
central value
high value
3,125
6,250
12,500
3,125 4,419 6,250 8,839
12,500
0,320 0,226 0,160 0,113 0,080
Long waves (LW)
low limit
central value
high values
12,500
25,000
50,000
12,500 17,678 25,000 35,355 50,000
0,080 0,057 0,040 0,028 0,020
6.2.3 Filtering algorithm
The filters used to break the original longitudinal profile into the previously defined bands, should be carefully chosen in order to introduce as little distortion as possible in the filtered signals, a common technique in that view is to use by digital forward and reverse filtering associated with measured section extending beyond the profile which is to be assessed.
prEN 13036-5:2006 (E)
15
6.2.4 Wave band indices
6.2.4.1 General
Different indices can be used in order to characterise the different wave band filtered profiles. These indices are denoted WBij, where WB denotes the wave band, i and j are the profile sample numbers corresponding respectively to the beginning and the end of the reporting length (for instance LW1024,2048, is the long waves Root Mean Square of the pre-processed profile elevations for the reporting length containing profile samples 1024 to 2048)
6.2.4.2 Root Mean Square value of the pre-processed profile elevations per wave band
The Root Mean Square value per wave band of the pre-processed profile elevations over a section Sij, where i and j are respectively the indices of the first and last point of the profile to be considered, is defined as:
∑=
=+−=
jk
ikz
ijWB 2
11
kwb,ij (4)
The index per wave band WBij can determined for each of the three wave band limited profiles zwb, by applying the above formula to the digital files obtained by filtering the pre-processed profile using the by the bi-octave filters described above.
These formulas applied to the three wavelength ranges give the indices:
SWij: short waveband (0,78125 m to 3,125 m) Root Mean Square value :
∑=
=+−=
jk
ikz
ijSW 2
11
ksw,ij (5)
MWij : medium waveband (3,125 m to 12,5 m) Root Mean Square value :
∑=
=+−=
jk
ikz
ijMW 2
11
kmw,ij (6)
LWij: long waveband (12,5 m to 50 m) Root Mean Square value :
∑=
=+−=
jk
iklz
ijLW 2
11
kw,ij (7)
The significance and values of the different WBij depend on the reporting length, in a non linear way, it is strongly recommended to use the following values:
100 m and at least 20 m for the SWij values,
100 m and at least 50 m for the MWij values,
at least 100 m for the LWij values.
prEN 13036-5:2006 (E)
16
6.3 PSD analyses
Assuming the road profile is stationary, different spectral estimation techniques can be applied. While these methods differ on seemingly minor points in their ways to compute the spectrum, they may give different results when it comes to estimating the spectrum which implies a major information reduction. It is therefore necessary to describe in a non ambiguous way and report with the results each step involved in a PSD computation.
The computation of PSD indices involves five major steps ( see Figure 6 )
6.3.1 Segmentation and windowing
In order to be able to report PSD indices for each part of the profile, the profile must be detrended and segmented into shorter segments the length of which should be the reporting length, these segments can be overlapped. Furthermore in order to minimise the side effects (called artefacts) this segmentation has on the computation of PSD, these segments must be windowed using a windowing (or smoothing) function. The overlapping method if used, as well as the windowing function must be clearly described.
6.3.2 PSD Computation
As the PSD is a defined as a limit, it is a theoretical value, it can be computed in many ways (for instance by squaring and normalising the spectrum) giving results which may be significantly different. The algorithm used must be fully described either as a full mathematical description or as a fully documented source of the program used to compute the PSD.
6.3.3 De-colouring
A profilometer is a filter which can emphasise or attenuate certain wavelengths, as such it can be described by a transfer function. This filtering effect has an influence on the PSD, de-colouring is a process which allows to minimise the alterations of the PSD. This is usually done by dividing the PSD by the squared transfer function modulus, but alternate methods such as de-colouring the profile itself can be used. The de-colouring method used must be fully described either as a full mathematical description or as a fully documented source of the program used to compute it.
6.3.4 Smoothing
The PSD curve is usually very angular or jagged ; excepted for very special applications, this tends to blur the overall aspect of the PSD graph with no advantage, a smoothing process is therefore used to reduce the visual clutter of the resulting graph as well as the number of points to take into account (for instance by smoothing the frequency bands). The smoothing method used must be fully described either as a full mathematical description or as a fully documented source of the program used to compute it.
6.3.5 Fitting and computation
The usual way to characterise a PSD is to find a mathematical curve which is close enough to it, this process it called fitting. Fitting is often carried out using a straight regression line characterised by one its point and its slope, other methods can be used. The fitting and computation method used must be fully described either as a full mathematical description or as a fully documented source of the program used to compute it.
prEN 13036-5:2006 (E)
17
Figure 6 — Computation of PSD indices
prEN 13036-5:2006 (E)
18
7 Test report
The following information must be included :
a) date of measurement;
b) location and identification of the road section;
c) description of the type of road measured;
d) weather conditions (if they may have an influence on the results);
e) average Annual Daily Traffic (AADT) and pavement age;
f) profile measurement length;
g) evaluation length;
h) class of the profilometer used (according to prEN 13036-6) for the vertical elevation, travelled distance accuracy, sampling interval, larger wavelength cutoff and reporting sampling interval;
i) date of the last calibration of the used profilometer and calibration report;
j) any parameter which may have an influence on the results, such as vehicle load and load repartition, operating speed;
k) raw or (preferably) re-sampled profile;
l) detailed description of the analyses performed (including reporting of the used algorithms and parameters);
m) results of the analyses.
prEN 13036-5:2006 (E)
19
Annex A (informative)
Re-sampling
NOTE The following text is neither intended to be a complete demonstration of the theoretical aspects of the re-sampling process which is beyond the scope of this document, but to provide the reader with some insights on how such a thing is feasible.
The Fourier theory proves that under certain assumptions, which are always satisfied in the real world of civil engineering, any function can be transformed into a finite sum of sine.
The Shannon theory proves that a given continuous signal, such as a sine, can be adequately (meaning that no information is lost or added in the process) described by a discrete set of points, obtained by sampling the sine at equally spaced steps, provided the sampling frequency (also called the sampling rate) is at least two times higher than the frequency of the sine (this condition is usually referred to as the Nyquist criterion).
The Shannon theory also proves that, once the condition on sampling frequency is met, no more information can be obtained through the use of a higher sampling frequency, conversely it proves that if the sampling frequency is lower than needed the sampled signal is a distorted representation of the original one, this is usually called aliasing. To prevent the occurrence of aliasing, one has to filter the signal in order to be sure that no frequency above half the sampling frequency is left in it.
Finally the Shannon theory proves that, once the condition on sampling frequency is met, it is possible to reconstruct the signal as it originally was using an interpolation procedure. This interpolation, which make use
of sync (x) = x
x)sin( functions, is very different from the linear interpolation which tends to spring to the mind
when one think about interpolation.
To sum up, the Shannon and Fourier theories prove, that provided the condition on sampling frequency is met, is possible to compute a perfect reconstruction of the original continuous signal from a discrete set of points. When the signal has been reconstructed, it is clear that it is possible to resample it at a higher frequency than the original one in order to be sure that the condition on sampling frequency will still be met.
The following description, while not aimed at demonstrating how the Shannon interpolation is carried out, is an attempt at shedding some light on the ideas on which it is based :
The Fourier Transform of a single sine wave, which meets the Nyquist criterion, is a complex number, which represent the frequency and phase of the sine wave. The correspondence between the sine wave and the aforementioned is bi-univocal: this means that giving the original sine equation (called its time domain representation) or the complex number (called its frequency domain representation) is from a mathematical point of view strictly equivalent.
Given that a sine is a function the value of which can be computed for any x, when its frequency and phase are known it follows that if it meets the Nyquist criterion, it is possible to add as many points to its spatial representation as one wants by using the Fourier Transform to find its frequency and phase (it is clear that for the Signal Processing theory these points neither add nor remove any information, or change the sine wave in any way).
The Fourier Transform is a linear one, that is if f(x) and g(x) are two functions and A and B two constants : FT( A*f(x) + B*g(x) ) = A*FT(f(x)) + B*FT(g(x)). This result can be extended to any finite number of functions.
Provided the original signal has been sampled in a way which meets the Nyquist criterion, it can be described as a finite sum of sine waves multiplied by constants.
It follows that provided the original signal has been sampled in a way which meets the Nyquist criterion, each of these sine waves can be re-sampled neither adding or removing any information.
prEN 13036-5:2006 (E)
20
Annex B (informative)
Wave bands analysis
B.1 Re-sampling and three bands filtering
This process can be applied to each reporting length or preferably to the whole profile measurement length (in order to limit filtering induced edge effects).
prEN 13036-5:2006 (E)
21
Figure B.1 — Re-sampling and three bands filtering process flowchart
prEN 13036-5:2006 (E)
22
B.1.1 Re-sampling and three bands filtering process detailed algorithm
NOTE 1 A SCILAB program designed to perform the re-sampling and filtering of the profile will be available from a ftp site at LCPC (France).
NOTE 2 All the filter descriptions follow the usual signal processing conventions [11], [12], where a filter response is characterised by its normalised transfer function defined as :
n
n
)()()()()()()()()(
)()()(
−−−−
−−−−
++++++++++++==
znzzzznbzbzbzbb
zAzBzH
1432114321
321
321
aaaa L
L (B.1)
Profile detrending: The profile is detrended using a linear regression, to remove the offset and DC component of the signal.
Resampling: The profile is re-sampled by
Applying a FIR (Finite Impulse Response) low pass filter to the original profile, using a forward and reverse filtering to prevent phase problems,
Computing the re-sampling ratio rounded to four digits after the decimal point,
Re-sampling using a polyphase resampling algorithm.
General lowpass filter: The profile is lowpass filtered using a Chebyshev type II filter of the 16th order with a 48 dB attenuation, the coefficients of which are given in the following table:
Table B.1
Constant z–1 z–2 z–3 z–4 z–5 z–6 z–7
b 0,0046 –0,0420 0,1858 –0,5260 1,0691 –1,6717 2,1324 –2,3636
a 1 –10,757 54,9259 –176,4702 398,8823 –672,0078 872,270 –889,284
z–8 z–9 z–10 z–11 z–12 z–13 z–14 z–15 z–16
2,4229 –2,3636 2,1324 –1,6717 1,0691 –0,5260 0,1858 –0,0420 0,0046
719,2920 –462,9043 236,1456 –94,4558 29,0319 –6,6269 1,0593 –0,1060 0,0050
A double forward and reverse filtering procedure is applied to prevent phase problems.
General highpass filter: The profile is highpass filtered using a Cauer type filter of the 5th order with a 130 dB attenuation and 0,01 dB ripple in the passband, the coefficients of which are given in the following table :
Table B.2
Constant z–1 z–2 z–3 z–4 z–5
b 0,9907 –4,9534 9,9067 –9,9067 4,9534 –0,9907
a 1 –4,9813 9,9252 –9,8881 4,9256 –0,9814
A double forward and reverse filtering procedure is applied to prevent phase problems.
Short waves highpass filter: The profile is lowpass filtered using a Chebyshev type II filter of the 10th order with a 123 dB attenuation, the coefficients of which are given in the following table :
prEN 13036-5:2006 (E)
23
Table B.3
Constant z–1 z–2 z–3 z–4 z–5 z–6 z–7 z–8 z–9 z–10
b 0,7624 -7,6211 34,2841 -91,4034 159,9341 -191,9122 159,9341 -91,4034 34,2841 -7,6211 0,7624
a 1 -9,4538 40,2361 -101,5249 168,1846 -191,1293 150,8993 -81,7277 29,0601 -6,1257 0,5813
A double forward and reverse filtering procedure is applied to prevent phase problems.
Long waves lowpass filter: The profile is lowpass filtered using a Chebyshev type II filter of the 7th order with a 48 dB attenuation, the coefficients of which are given in the following table :
Table B.4
Constant z–1 z–2 z–3 z–4 z–5 z–6 z–7
b 0,0005 –0,0024 0,0043 –0,0024 –0,0024 0,0043 –0,0024 0,0005
a 1 –6,8707 20,2323 –33,1012 32,4951 –19,1412 6,2643 –0,8787
A double forward and reverse filtering procedure is applied to prevent phase problems.
Medium waves bandpass filter: The profile is bandpass filtered using two Chebyshev type II filters :
the first one is a highpass filter of the 8th order with a 48 dB attenuation, the coefficients of which are given in the following table:
Table B.5
Constant z–1 z–2 z–3 z–4 z–5 z–6 z–7 z–8
b 0,0034 –0,0253 0,0839 –0,1626 0,2010 –0,1626 0,0839 –0,0253 0,0034
a 1 –7,3909 23,9205 –44,2781 51,2695 –38,0249 17,6403 –4,6800 0,5436
the second one is a lowpass filter of the 6th order with a 24 dB attenuation, the coefficients of which are given in the following table:
Table B.6
Constant z–1 z–2 z–3 z–4 z–5 z–6
b 0,9767 –5,8593 14,6474 –19,5295 14,6474 –5,8593 0,9767
a 1 –5,9521 14,7625 –19,5284 14,5318 –5,7676 0,9538
A double forward and reverse filtering procedure is applied for each filter to prevent phase problems.
prEN 13036-5:2006 (E)
24
Key 1 filter responses 2 wave length
Figure B.2 — 3-bands filter response curves
B.2 Detailed characteristics of filters
Table B.7 — Amplitude responses of the CEN filters
Spatial filtering in wavelengths Band Bi-octave
m Octave
m
Spatial frequencies in cycles per metre
Short waves (SW)
low limit
central value
high limit
0,78125
1,5625
3,125
0,78125
1,1048543456039800
1,5625
2,2097086912079600
3,1250
1,280
0,9050966799187810
0,640
0,4525483399593900
0,320
prEN 13036-5:2006 (E)
25
Medium waves (MW)
low limit
Central value
high limit
3,125
6,250
12,500
3,125
4,4194173824159200
6,250
8,8388347648318400
12,500
0,320
0,2262741699796950
0,160
0,1131370849898480
0,080
Long waves (LW)
low limit
central value
high limit
12,500
25,000
50,000
12,500
17,6776695296637000
25,000
35,3553390593274000
50,000
0,080
0,0565685424949238
0,040
0,0282842712474619
0,020
prEN 13036-5:2006 (E)
26
Annex C (informative)
An illustration of PSD computation
C.1 General
This annex is an example based on the PSD computation as used in Germany and defined in the following documents: "Zusätzliche Technische Vertragsbedingungen, Technische Erfassungs- und Auswerteregeln, Teil D" and "Zusätzliche Technische Vertragsbedingungen zur Zustandserfassung und -bewertung von Straßen (ZTV ZEB-StB)" is intended to show how an already existing way of computing the PSD can be described within the frame given under clause 7.
C.2 Segmentation and windowing
The PSD computations are based on non overlapping sections, the length of which is 102,4 m (corresponding to N = 1 024 samples).
The profiles used are sampled at the fixed spatial sampling interval of δ x = 0,1 m; obtained by decimating the standard re-sampled profile by a factor of 2, and filtering the resulting profile with a moving average filter of length N' (where N' = 1 023 ) using the following equation:
∑
−′=
−′=
+′−=
21
21
1
Nj
Nj
zN
zZ jiii (C.1)
NOTE To obtain the Moving Average in a range i1⋅δx ≤ i⋅δx ≤ i2⋅δx the profile must be processed within the range (i' indicates the range the of original profile): (i1 – (N' – 1)/2)⋅ δx ≤ i’⋅δx ≤ (i2 + (N' – 1)/2)⋅ δx.
The resulting profile is windowed using a Cosine Digital Tapering Window (CDTW) of length N = 1 024, defined by the following equations :
100for
252 NiN
iC <≤
−= ππcosi (C.2)
109
10for1 NiNC ≤≤=i (C.3)
NiNN
iC <<
−=
109for
2952 ππcosi (C.4)
To calculate the PSD for a section with length L, the position of the N profile displacements for input processing shall be
Zi : = Ci-k Zi-k with k = (N – L)/2 – 1 (C.5)
prEN 13036-5:2006 (E)
27
C.3 PSD Computation
The PSD computation is based on the Digital Complex Fourier Transform defined by the following equation :
110with1
0
2−== ∑
−=
=
−
NkeZZNj
j
Nkji
,,,jk Lπ
(C.6)
The raw PSD is computed by normalising the squared modulus of the Fourier Transform using the following equation:
xncZ
NncG
δπ 21and8750where
21 2
=== maxkmax
*k , (C.7)
The correction factor 1/c being used to compensate for the CDTW amplitude reduction.
C.4 De-colouring
The raw PSD is divided by the square of the modulus of the transfer function V(nk) according to the following equation (the transfer function V(nk) depends on the individual transfer characteristic of the used device: the transfer function influences the characteristic of the delivered longitudinal profile data) :
( ) 2k
*k
knV
GG = (C.8)
C.5 Smoothing
The decoloured PSD is first smoothed by a three-point smoothing process to provide a reduction of the random error.
211 5050 GGG ′×+′×= ,, (C.9)
12
2with25050250 11 −=′×+′×+×= +−NkGGGG ,,,,, kkkk L (C.10)
2122 5050 N/N/N/ ,, GGG ′×+′×= − (C.11)
NOTE The spectral data G'0 is not used for further calculations.
The three-point smoothed PSD is then smoothed by using the mean value of this PSD in the following frequency bands:
octave bands from the lowest frequency (zero excepted) up to a centre frequency of 0,0312 cycles/m (corresponding to a wavelength of 32,05 m),
third-octave bands from the last octave band up to a centre frequency of 0,25 cycles/m (corresponding to a wavelength of 4 m),
sixth-octave bands from the last third-octave band up to a center frequency of 1 cycle/m (corresponding to a wavelength of 1 m)
twelth-octave bands from the last sixth-octave band up to a the highest calculated frequency of 6 cycles/m (corresponding to a wavelength of 0,167 m).
prEN 13036-5:2006 (E)
28
The centre frequencies to be used for the calculation of the smoothed PSD are given in Table C.1.
NOTE nl = lower cut-off frequency, nc = centre frequency, nh = upper cut-off frequency.
A small overlap exists between the lowest twelfth-octave bandwidth and the highest third-octave bandwidth. This overlap maintains the values 0,5; 1; 2; 4 as centre frequencies in the twelfth-octave bands. This makes it convenient to calculate the road characterisation immediately from the twelfth-octave band smoothing.
The lower cut-off nl, centre nc and upper cut-off nh frequencies can be derived from:
nl := 2EXP–B/2 m–1, nc := 2 EXP m–1, nh := 2 EXP+B/2 m–1
where EXP is the exponent and B the width of the band:
Octave bandwidth B = 1; EXP = –9, –8,…, -5;
Third-octave bandwidth B = 1/3; EXP = –4 1/3, –4,…, –2;
Sixth-octave bandwidth B = 1/6; EXP = –1 5/6, –1 2/3, …, 0;
Twelfth-octave bandwidth B = 1/12; EXP = 1/12, 1/6, … .
prEN 13036-5:2006 (E)
29
Table C.1 — Centre frequencies and cut-off frequencies for PSD smoothing, expressed in spatial frequency, n
a) Octave bandwidth d) Twelfth-octave bandwidth
EXP nl
m–1
nc
m–1
nh
m–1
EXP nl
m–1
nc
m–1
nh
m–1
–9 0,001 4 0,002 0 0,002 8
–8 0,002 8 0,003 9 0,005 5
–7 0,005 5 0,007 8 0,011 0
–6 0,011 0 0,015 6 0,022 1
–5 0,022 1 0,031 2 0,044 2
b) Third-octave bandwidth
EXP nl
m–1
nc
m–1
nh
m–1
–4,333 0,044 2 0,049 6 0,055 7
–4 0,055 7 0,062 5 0,070 2
–3,667 0,070 2 0,078 7 0,088 4
–3,333 0,088 4 0,099 2 0,111 4
–3 0,111 4 0,125 0 0,140 3
–2,667 0,140 3 0,157 5 0,176 8
–2,333 0,176 8 0,198 4 0,222 7
–2 0,222 7 0,250 0 0,280 6
c) Sixth-octave bandwidth
EXP nl
m–1
nc
m–1
nh
m–1
–1,833 0,2649 0,2806 0,2973
–1,667 0,2973 0,3150 0,3337
–1,500 0,3337 0,3536 0,3745
–1,333 0,3745 0,3969 0,4205
–1,167 0,4205 0,4454 0,4719
–1 0,4719 0,5 0,5297
–0,833 0,5297 0,5612 0,5946
–0,667 0,5946 0,6300 0,6674
–0,500 0,6674 0,7071 0,7492
–0,333 0,7492 0,7937 0,8409
–0,167 0,8409 0,8909 0,9439
0 0,9439 1 1,0596
0,083
0,167
0,250
0,333
0,417
0,500
0,583
0,667
0,750
0,833
0,917
1
1,083
1,167
1,250
1,333
1,417
1,500
1,583
1,667
1,750
1,833
1,917
2
2,083
2,167
2,250
2,333
2,417
2,500
2,583
2,667
2,750
2,633
2,917
3
1,029 3
1,090 5
1,155 4
1,224 1
1,296 8
1,374 0
1,455 7
1,542 2
1,633 9
1,731 1
1,834 0
1,943 1
2,058 6
2,181 0
2,310 7
2,448 1
2,593 7
2,747 9
2,911 3
3,084 4
3,267 8
3,462 1
3,668 0
3,886 1
4,117 2
4,362 0
4,621 4
4,896 2
5,187 4
5,495 8
5,822 6
6,168 8
6,535 7
6,024 8
7,336 0
7,772 3
1,059 5
1,122 5
1,189 2
1,259 9
1,334 8
1,414 2
1,498 3
1,587 4
1,681 8
1,781 8
1,887 7
2
2,118 9
2,244 9
2,378 4
2,519 8
2,669 7
2,828 4
2,996 6
3,174 8
3,363 6
3,563 6
3,775 5
4
4,237 9
4,489 8
4,756 8
5,039 7
5,339 4
5,656 9
5,993 2
6,349 6
6,727 2
7,127 2
7,551 0
8
1,090 5
1,155 4
1,224 1
1,296 8
1,374 0
1,455 7
1,542 2
1,633 9
1,731 1
1,834 0
1,943 1
2,058 6
2,181 0
2,310 7
2,448 1
2,593 7
2,747 9
2,911 3
3,084 4
3,267 8
3,462 1
3,668 0
3,886 1
4,117 2
4,362 0
4,621 4
4,896 2
5,187 4
5,495 8
5,822 6
6,168 8
6,535 7
6,924 3
7,336 0
7,772 3
8,234 4
The mean PSD in a defined band should be calculated in the following way:
prEN 13036-5:2006 (E)
30
for nH > nL + 1 (more than two considered PSD frequency components in band i) is
(1) (2) (3)
( )[ ] [ ])(
)()(),()(
)(
)(
)()()(
)(,)( H
lh
eHh
)l(h
e
Llh
leLs
H
L nGinin
Bninnin
BjG
nGinin
inBniG
i
n
nj ×−
×−−+
−
×
+×−
−×+=
∑−
+= 5050
1
1 (C.12)
where
Gs(i) is the smoothed PSD in smoothing band;
+= 50,
)(
e
hH B
inINTn , (nh: see Table 4); (C.13)
+= 50,
)(
e
lL B
inINTn (nl: see Table 4); (C.14)
nL and nH are the numbers of first and last considered PSD frequency component in actual band.
The first and the third term of the right side of the equation calculate respectively the parts of the original band nl, and nh in the calculated smoothed band i.
The smoothed form of the PSD may be fitted by a straight line on the smoothed data by the least-square-method in the spatial frequency range 0,011 cycles/m to 2,83 cycles/m.
NOTE Calculation in special cases:
— for nH = nL is Gds (i) = Gd (nH = nL), (only one considered frequency component in band);
— for nH = nL + 1 calculate only term (1) and (3), (only two considered frequency components in band);
— for the last upper band (if nh (i) > (nmax + 0,5) Be ): calculate only term (1) and (2) with nh (i) = (nmax + 0,5) Be and summation in term (2) from j = nL + 1 to nmax , where nmax is the highest considered frequency component in smoothing process.
C.6 Fitting and computation
The PSD is reported on a graph using a loglog scale (where the X axis represents the spatial frequencies in cycles/m and the Y axis represents the PSD in m³) where both the smoothed PSD and the fitted PSD are shown.
The fitted PSD is obtained by fitting a straight line by the least-mean-square method in the spatial frequencies range 0,011 cycles/m to 2.83 cycles/m (0,063 rad/m to 17,77 rad/m), the equation of such a fitted curve is given by the following equation :
w
nnnGnG
−
×=
00 )()( where n0 = 0,1 cycles/m is the reference spatial frequency (C.15)
or
prEN 13036-5:2006 (E)
31
w
GG−
ΩΩ×Ω=Ω
00 )()( where Ω0 = 1 rad/m is the reference angular spatial frequency (C.16)
where
G(Ω0) is the unevenness index;
w is a measure of the waviness of the spectrum.
The PSD is characterised by the values of G(Ω0) and w. The two following pages give an illustration of the smoothing process, they are issued from ISO 8086.
prEN 13036-5:2006 (E)
32
Key 1 displacement power spectral density, in cubic metre (m³) 2 wave length λ, in metre (m)
3 spatial frequency n in cycles per metre 4 angular spatial frequency Ω, in rad per metre
Figure C.1 — Non-smoothes PSD
prEN 13036-5:2006 (E)
33
Key 1 displacement power spectral density, in cubic metre (m³)
2 wave length λ, in metre (m)
3 spatial frequency n in cycles per metre
4 angular spatial frequency Ω, in rad per metre
Figure C.2 — Smoothed PSD
prEN 13036-5:2006 (E)
34
Country: Belgium
Road: N 1000
Place: Xstad
Direction: North to South
Concrete pavement
Distance track to right roadside: 1 m
Distance travelled: 3 571 m
Be = 0,005 5 cycles/m
ε r = 0,23
Statistical precision = ± 44 %
Genetral characterization: 0,011 < n < 2,83; rmsd = 0,038 4 m
Linear fitting: w = 2,16; Gd(0,1) = 892 ×10–6 m³
prEN 13036-5:2006 (E)
35
Bibliography
[1] Sayers, M.W., T.D. Gillespie, and Querioz.C.A.V., International Experiment to Establish Correlations and Standard Calibration Methods for Road Roughness Measurements. World Bank Technical Paper 45. World Bank, Washington, D.C., 1986
[2] Sayers, M.W., T.D. Gillespie, and W.D., Paterson Guidelines for the conduct and calibration of Road Roughness Measurements. World Bank Technical Paper 46. World Bank, Washington, D.C., 1986
[3] Karamihas S.M., and T.D. Gillespie et al. “Guidelines for Longitudinal Pavement Profile Measurement”, TRB - NCHRP Report 434, Washington D.C., 1999,
[4] Sayers, M.W., “On the calculation of International Roughness Index from Longitudinal Road Profile” Transportation Research Record 1501, Transportation Research Board, pp. 1-12”, Washington D.C., 1995
[5] ASTM, “Computing International Roughness Index of Roads from Longitudinal Profile measurements”, E1926-98, ASTM Standards
[6] Descornet, G: “Inventory of High-Speed Longitudinal and Transverse Road Evenness Measuring Equipment in Europe”, BRRC editor. FEHRL Technical Note 1999/01, Brussels 1999.
[7] Willet M, Magnusson G., Ferne B., “FILTER- Theoretical study of Indices”, FEHRL Technical Note 2000/2
[8] de Witt, Kempkens E, Sjögren L., Ducros D.M.,”The FILTER Experiment”, FEHRL Technical Note 1999/2
[9] Ducros D.M.,Petkovic L., Descornet G., Berlémont B., Alonso M., Yanguas S., Jendryka W., Andrén P, “FILTER Experiment – Longitudinal Analyses”, Final Report 2001/1, FEHRL
[10] International Experiment to Harmonize Longitudinal and Transverse Profile Measurement and Reporting Procedure (The EVEN Project), PIARC Technical report 01.07.B, 2002
[11] Oppenheim A.V., Schafer R. W., “Digital Signal Processing”, Prentice Hall inc. editor, Englewood Cliffs, New Jersey, USA, 1975
[12] Feuer A., Goodwin G.C., “Sampling in Digital Signal Processing and Control, Birkhäuser editor, Boston, USA, 1996
