ISO/TR 8955:2025

Technical
Report
ISO/TR 8955
First edition
Railway infrastructure — Track
2025-03
quality evaluation — Chord-
based method
Applications ferroviaires — Qualité géométrique de la voie —
Méthode sur la base du cordeau
Reference number
© ISO 2025
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ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms. 2
5 Relationship of measuring systems and evaluation methods for longitudinal level and
alignment . 3
5.1 Measuring systems .3
5.2 Evaluation methods .4
5.3 Relationship of measuring systems and evaluation methods .4
6 Chord measurement system and chord-based method of longitudinal level and
alignment . 5
6.1 Chord measurement system . .5
6.1.1 General .5
6.1.2 Symmetrical chord method .5
6.1.3 Asymmetrical chord method .5
6.2 Transfer function .6
6.2.1 General .6
6.2.2 Ratio of the chord division .7
6.2.3 Chord length .10
6.3 Chord based method and chord length class .10
6.3.1 Chord based method .10
6.3.2 Chord length class .11
7 Colouring process .11
7.1 Definition of colouring process .11
7.2 Colouring method . 12
7.2.1 Digital filtering by FIR (finite impulse response) . 12
7.2.2 Digital filtering by IIR (infinite impulse response) .14
7.2.3 Space domain method .14
7.2.4 Frequency domain method .14
7.3 Verification of colouring process . 15
7.3.1 Introduction . 15
7.3.2 Verification with test signals .16
7.3.3 Verification with recorded track geometry data .17
8 Recolouring process .18
8.1 Definition of recolouring process .18
8.2 Recolouring method . 20
8.3 Verification of recolouring process . 20
9 Elimination of influence of track geometry layout.20
10 Usage of restored waveform .21
Annex A (informative) Example of colouring method in MATLAB language .22
Annex B (informative) Verification of colouring/decolouring .30
Bibliography .43

iii
Foreword
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complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
ISO 23054-1 describes the wavelength range method and the chord-based method in parallel as methods
for evaluating track geometry quality. ISO 23054-1 specifies a detailed technique for obtaining the track
geometry in a desired wavelength range from the signal measured by an inertial measurement system
or the chord measurement system. ISO 23054-1 does not specify a technique for obtaining the signal by a
particular chord-based method from the signal measured by an inertial measurement system or a chord
measurement system with a different chord base.
The chord-based method is a method of managing the track geometry using an evaluation signal that
emphasizes the track geometry of a specific wavelength component, and is a method used in many countries.
The chord measurement system and the chord-based method are used to measure and evaluate the track
geometry parameters: alignment and longitudinal level.
This document provides information on the chord-based method for track geometry evaluation, looking
specifically at the relationship between an inertial measurement system and a chord measurement system
applying the chord-based method with different chord length and chord division. This document is intended
to be used in conjunction with ISO 23054-1.

v
Technical Report ISO/TR 8955:2025(en)
Railway infrastructure — Track quality evaluation — Chord-
based method
1 Scope
This document describes the relationship between an inertial measurement system and a chord
measurement system applying the chord-based method with different chord length and chord division.
This document is applicable to 1 435 mm and wider track gauges. This document does not apply to urban/
light rail systems, tramways and any track gauge narrower than 1 435 mm.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content constitutes
requirements 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.
ISO 23054-1, Railway applications — Track geometry quality — Part 1: Characterization of track geometry and
track geometry quality
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 23054-1 and the following apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
wavelength range method
evaluation method for longitudinal level and alignment using signals not coloured by the transfer function
3.2
chord-based method
evaluation method for longitudinal level and alignment using signals coloured by the transfer function
3.3
coloured signal
signal distorted in magnitude and phase by the transfer function
3.4
colouring
process of generating a chord measurement signal from a non-coloured measurement signal
3.5
decolouring
process of removing the distortion of a chord measurement signal to retrieve a non-coloured
measurement signal
3.6
recolouring
process of transforming a coloured signal into another coloured signal

4 Symbols and abbreviated terms
For the purposes of this document, the symbols given in ISO 23054-1 and following apply.
a length of trailing chord section (i.e. L = a + b), variations: a , a
e m
a filter coefficient (denominator) of the Z-transform
k
b length of trailing chord section (i.e. L = a + b) , variations: b , b
e m
b filter coefficient (numerator) of the Z-transform
k
d distance (m)
d sampling distance for variable s (m)
s
C short chord length class based on line speed
C long chord length class based on line speed
F Fourier Transform
−1
F inverse of the Fourier Transform
h(s) impulse response
h (q) discrete form of h such that h (q) = h(q·d )
s s s
w
H
transfer function in Fournier domain, Fournier transform of h(s), variations: H , H , H , H , H , H
I R e em m m
H colouring transfer function for chord [a , b ]
e e e
H transfer function between the measurement signal and the evaluation signal (i.e. the transfer function
em
for the recolouring process)
H imaginary part of the transfer function
I
H colouring transfer function for chord [a , b ]
m m m
H real part of the transfer function
R
w
decolouring transfer function for chord [a , b ] (i.e. the Wiener inverse of H )
H m m m
m
H (p) transfer function of h in the Laplace domain
L
E
H estimation of H
H (Z) transfer function in the form in the Z-domain
z
j imaginary unit
l normalized chord length
L chord length
m magnitude of the transfer function
M magnitude of the coefficient of Z-transform (numerator and denominator)
p Laplace variable
s
q
discrete form of the space variable s (i.e. q= ; q is an integer)
d
s
r chord division ratio
R Noise over signal ratio
n/s
s space variable (m)
S power spectral density of the input signal x(s)
XX
S cross-spectral density of the input signal x(s) and the output signal y(s)
YX
V offset from chord to rail at the measurement point
w wavelength (m)
W wavelength range (see ISO 23054-1)
x
x (q) discrete form of signal x such that x (q) = x(q·d )
s s s
x(s) variable input signal along space, e.g. versine signal
X real axis
r
X(ω) Fourier Transform of x(s)
y (s) amplitude of y(s) (m)
a
y (s) amplitude error of an alignment or longitudinal level signal (m)
e
y(s) variable output signal along space, e.g. alignment signal or longitudinal level signal
Y imaginary axis
i
y (q) discrete form of signal y such that y (q) = y(q·d )
s s s
D D
y (s) space shifted version of y(s) with delay D y (s) = y(s+D)
D D D
y (q) discrete form of signal y (s) such that y (q) = y(q·d + D)
s s s
Y(ω) Fourier Transform of y(s)
δ Dirac delta function
ω spatial angular frequency
5 Relationship of measuring systems and evaluation methods for longitudinal level
and alignment
5.1 Measuring systems
There are two measuring systems used for longitudinal level and alignment:
— inertial measurement system;
— chord measurement system.
5.2 Evaluation methods
There are two evaluation methods for longitudinal level and alignment:
— wavelength range method;
— chord-based method.
5.3 Relationship of measuring systems and evaluation methods
Measuring systems and evaluation methods of track geometry have a relationship which permits mutual
conversion between them (Figure 1).
Key
1 inertial measurement system
2 chord measurement system
3 wavelength range method
4 chord based method
a
Filtering process to convert track geometry measured by an inertial measurement system for the evaluation of track
irregularity by wavelength range method; see ISO 23054-1:2022, Annex C and D.
b
Colouring process to convert track geometry measured by an inertial measurement system for the evaluation of
track irregularity by chord-based method, see Clause 7.
c
Decolouring process to convert track geometry measured by a chord measurement system for the evaluation of track
irregularity by wavelength range method; see ISO 23054-1:2022, Annex E.
d
Recolouring process to convert track geometry measured by a specified chord measurement system for the
evaluation of track irregularity by another chord-based method; see Clause 8. If the same chord for measurement
and assessment is used, the recolouring process is not used.
Figure 1 — Measuring systems and evaluation methods relationship
The railway authority and infrastructure manager can adopt any of the measuring systems and evaluation
methods listed in this clause.

6 Chord measurement system and chord-based method of longitudinal level and
alignment
6.1 Chord measurement system
6.1.1 General
The chord measurement system is a method for measuring the longitudinal level and alignment of the track
geometry. The chord measurement system measures the offset from a straight chord to the rail at a defined
position between the two end points of the chord.
The straight chord can be either mechanical or optical. In both cases, the interaction between the static and
dynamic behaviour of the vehicle (or the carrier of the measurement system) is evaluated with regards to
the required accuracy.
NOTE Chord measurement systems have come to be applied to vehicles recording track geometry. The body
of the vehicles is used as the reference chord for measurement. As the deformation of the car body cannot always
be neglected, a laser beam installed inside the car body of the track geometry recording vehicles can be used as a
reference chord.
6.1.2 Symmetrical chord method
In the case of a symmetrical chord, the offset from the rail is measured in the centre of the chord. This
means that the lengths of both chord sections are not equal; see Figure 2.
Key
1 rail
2 chord
3 measurement direction
V offset from chord to rail at the measurement point
L chord length
a divided chord length, L = 2a
Figure 2 — Symmetrical chord method
6.1.3 Asymmetrical chord method
In the case of an asymmetrical chord, the offset from the rail is not measured in the centre of the chord. This
means that the lengths of both chord sections are not equal; see Figure 3.

Key
1 rail
2 chord
3 measurement direction
V offset from chord to rail at the measurement point
L chord length
a, b divided chord length, L = a + b
Figure 3 — Asymmetrical chord method
6.2 Transfer function
6.2.1 General
The transfer function between the alignment or the longitudinal level and the signal obtained with a chord
measurement system (L = a + b) is expressed by Formulae (1) and (2). The absolute value of the transfer
function |H(λ)| indicates the amplitude characteristic, and the argument of the transfer function ∠ H(λ)
indicates the phase characteristic.
As the chord division order, regarding the measurement direction, is substantial for the transfer function
expression, in this document "a" is defined as the trailing chord division and "b" is defined as the leading
chord division.
2 2
 
b 22ππa a b b 2πa a 2πb
         
Hw() =−1 cosc− os + sin − sin (1)
 
         
L w L w L w L w
         
 
 
Hw()
−1
I
∠Hw =tan (2)
()
Hw()
R
where
H is the transfer function;
w is the wavelength;
L is the chord length;
a, b are the divided chord lengths, L = a + b;
NOTE 1  In the case of symmetrical chord, b = a.
NOTE 2  The order of a and b is meaningful; see Figure 3.
H is the imaginary part of the transfer function;
I
H is the real part of the transfer function.
R
The complete formulation of the transfer function can be found, for example, in Formulae (3) and (4). A
detailed presentation is provided in Reference [1]. The Laplace transform of the filter can be expressed as
follows:
b a
 
Hp()= expe()−ap + xp()bp −1 (3)
L
 
+ +
ab ab 
2π
ps=+ jsω=+ j (4)
w
where
H is the transfer function in the form in the Laplace domain;
L
p is the Laplace variable;
s is the distance;
j is the imaginary unit;
ω is the spatial angular frequency.

This continuous spectrum is obtained as described in Reference [1] from the impulse function response in s
in Formula (5):
b a
hs()= δδ()sa− + ()sb+ −δ ()s (5)
ab+ ab+
where
h is the impulse function;
s is the measurement distance;
δ is the Dirac delta function.
6.2.2 Ratio of the chord division
By changing the ratio of the chord division of the chord measurement system, the characteristics of the
transfer function between the alignment or the longitudinal level and the signal obtained with a chord
measurement system (L = a + b) also change.
Figure 4 shows an example transfer function with a chord length, L, of 10 m and a chord division (a = b) of
5 m. This method is called the 10 m symmetrical chord method. In the symmetrical chord method, the gain
is 2 at wavelength L/(2n-1) with n ≥ 1. The gain is zero at wavelength L/(2n) with n ≥ 1, where n is a positive
integer.
Key
m magnitude of the transfer function
w wavelength (m)
1 transfer function — 5 m/5 m
Figure 4 — Example of transfer function of 10 m symmetrical chord method (a = b = 5 m)
To avoid these zero gains in the transfer function, it is advisable to set the ratio of chord division to a
condition that is not an integral multiple. Figure 5 shows the transfer functions for the case where:
— L = 10 m, a = 2,5 m and b = 7,5 m (b = 3a); and
— L = 10 m, a = 4,5 m and b = 5,5 m (b = (5,5/4,5)·a approximately 1,22·a).
These are called 10 m asymmetrical chord methods.
The transfer function has zero gain at each wavelength equal to a common divisor of the chord divisions.
For example, for L = 4 m + 6 m, the greatest common divisor (GCD) is 2 m and the zero gains occur at all
wavelengths, w = GCD/n = 2/n with n ≥ 1.

Key
m magnitude of the transfer function
w wavelength (m)
1 transfer function — 2,5 m/7,5 m
2 transfer function — 4,5 m/5,5 m
Figure 5 — Example of transfer function of 10 m asymmetrical chord method (Chord divided:
a = 2,5 m, b = 7,5 m, and a = 4,5 m, b = 5,5 m)
Figure 6 shows the amplitude characteristics of the transfer function depending on the chord division ratio.
The horizontal axis shows the normalized chord length. From the figure, the gain on the short wavelength
side is strongly affected by the chord division, but the gain on the long wavelength side is not so affected by
the chord division.
Key
m magnitude of the transfer function
l normalized wavelength w/L
r chord division ratio a/L
1 symmetrical chord
Figure 6 — Amplitude characteristics of transfer function due to chord division ratio

6.2.3 Chord length
When the chord length of the chord measurement system is changed, the transfer function between the
alignment/longitudinal level and the signal obtained with a chord measurement system (L = a + b) also
changes.
Longer chord lengths are suitable for assessing longer wavelengths.
Figure 7 shows the transfer functions of the 5 m symmetrical chord, the 10 m symmetrical chord, and the
40 m symmetrical chord as examples of the symmetrical chord method. In this figure, the transfer function
with a wavelength shorter than 1/2 of the chord length of each symmetrical chord method is not plotted.
Key
m magnitude of the transfer function
w wavelength (m)
1 transfer function — 5 m symmetrical chord
2 transfer function — 10 m symmetrical chord
3 transfer function — 40 m symmetrical chord
Figure 7 — Example of transfer function of symmetrical chord method with different chord length
6.3 Chord based method and chord length class
6.3.1 Chord based method
The evaluation of longitudinal level and alignment in a chord-based method is obtained by applying the
transfer function of the chord measurement system (shown in 6.2) to the real track geometry. As shown in
6.2.3, the chord-based method can emphasize the track geometry in a specific wavelength.
This method is used, for example, by using a 10 m chord to evaluate the short wavelength of track geometry
with regard to the running safety of railway vehicles. Similarly, with a 40 m chord, the chord-based method
is used to assess the long wavelength track geometry, as regards the riding comfort on high-speed lines.

6.3.2 Chord length class
In ISO 23054-1, C and C are shown as chord length classes (Table 1). C represents values of 5 m – 20 m and
1 2 1
C represents values longer than 20 m. In addition, C is generally considered for line speed of 250 km/h or
2 2
greater.
Table 1 — Chord length class
Chord length class Base chord
m
C 5 – 20
C > 20
7 Colouring process
7.1 Definition of colouring process
Colouring is a method to obtain the evaluation signal of the track geometry with an arbitrary chord length
(and chord length division) from the measurement data of the track geometry by an inertial measurement
system as shown in Figure 1. For example, it is used to convert the data measured by the track geometry
device of the inertial measurement system into the evaluation waveform of the 10 m chord methods.
Figure 8 shows an example of the deformation of a track geometry signal measured by an inertial
measurement system, when a 10 m chord-based filter both with symmetric and asymmetric division is
applied.
Key
1 inertial measured real shape signal (wavelength range: 3 m – 150 m)
2 coloured waveform (10 m symmetrical chord, 5 m/5 m)
3 coloured waveform (10 m asymmetrical chord, 2,5 m/7,5 m)
d distance (m)
y amplitude (mm)
a
Figure 8 — Example of distortion due to different chord division in colouring process

Figure 9 shows an example of the effect of the different chord lengths on the colouring process by comparing
signals obtained with:
— 20 m symmetrical chord method, and
— 40 m symmetrical chord method,
with regards to the same signal, W according to ISO 23054-1, on a short track section.
Key
1 inertial measured signal (wavelength range: 25 m – 70 m)
2 coloured waveform (20 m symmetrical chord, 10 m/10 m)
3 coloured waveform (40 m symmetrical chord, 20 m/20 m)
d distance (m)
y amplitude (mm)
a
Figure 9 — Example of distortion due to different chord lengths in colouring process
7.2 Colouring method
7.2.1 Digital filtering by FIR (finite impulse response)
7.2.1.1 Overview
This method consists of expressing the filter described in 6.2.1 as a Fourier transform.
7.2.1.2 Case 1
One way of doing this is by discretization of the Laplace transform in Formula (5) in a Z transform as shown
in Formula (6):
n
−k
HZ = bZ (6)
()
z k
∑
k=0
where
H is a transfer function in the form of Z-transform;
z
Z is a complex number;
n is a positive integer;
b is a filter coefficient.
k
Commonly, an approximation of this formula is obtained by estimation of the b coefficient with finite values
k
of n, applying optimization methods. It is usually easy to respect magnitude and phase response at a cost of
large values of n.
When n is small enough, an online data treatment is possible.
An example of simple FIR design in MATLAB language is given in Clause A.4.
7.2.1.3 Case 2
Another method is through discrete convolution of the impulse response h in a digital filter as shown in
Formula (7):
hk() = (hk⋅=ds)  kK01,,⋅⋅⋅, −1 (7)
s
where
h is the impulse response;
h Is the discrete form of h;
s
K is a positive integer.
K−1
A non-cyclic linear phase filter has a phase delay of α = d . To establish a good positional relationship
s
[2]
to the track, it is appropriate to use a signal rectified for the phase delay. In Japan, the LABOCS system is
used to obtain a coloured output signal without phase delay by applying the impulse response of Formula (8)
to the input signal. Examples of colouring process can be found in Figure 8 and Figure 9.
K−1
yn()=−hk()  xn()kn⋅= 01,,⋅⋅⋅⋅⋅,N−1 (8)
ss s
∑
k=0
where
y is the output data in discrete form;
s
x is the input data in discrete form;
s
N is the maximum number of data.
See Reference [2] for further information.
An example of simple discrete convolution in MATLAB language is given in Clause A.4.
Cases 1 and 2 are mathematically equivalent, since h(k) would be the discretization of the continuous
impulse response of h while H (Z) would be the Z transform of the bilateral Laplace transform.
c c
7.2.2 Digital filtering by IIR (infinite impulse response)
This method consists of expressing the filter described in 6.2.1 as a Fourier transform, in a Z transform as
shown in Formula (9):
n
−k
bZ
∑ k
k=0
HZ() = (9)
z
m
−k
1+ aZ
k
∑
k=1
Commonly an approximation of this formula is obtained by estimation of the ()ba, coefficient with finite
kk
and small values of (n, m). Finding the optimal coefficient can be quite complex; therefore, a combination of
analogue filters is usually used to apply gain at a selective waveband, resulting in nonlinear phase evolution.
An example of simple IIR design in MATLAB language is given in Clause A.4.
7.2.3 Space domain method
Since the transfer function is known, if treatment can be made offline, a simple but memory-intensive
method can be applied [see Formula (10)].
The input signal can be convoluted with the impulse response:
∞
ys =−hs tx tdt (10)
() () ()
∫
−∞
where
s is the distance:
t is an integration variable.
An example of the simple space domain method in MATLAB language is given in Clause A.3.
7.2.4 Frequency domain method
Since the transfer function is known, if treatment can be made offline, a simple but memory-intensive
method can be applied as follows [see Formula (11)].
1) The Fourier transform of the input signal is computed; thereby the input signal spectrum is obtained at
given frequencies.
2) Evaluate the filter value at each of those frequencies.
3) Multiply the input spectra and the filter values at the same frequency, resulting in the output signal
spectra.
4) Take the inverse Fourier transform of this spectrum to obtain the output signal.
XFω = xs
() ()()


b a
 

HH()ωω==()pj01+ = expe()−aj ωω+ xp()bj −
L
 
ab+ ab+
 
(11)


YH()ωω= ()*X ()ω

−1 
ys() =FY()()ω

where
s is the space variable (m);
x(s) is a variable input signal along space;
y(s) is a variable output signal along space;
X(ω) is the Fourier Transform of x(s);
Y(ω) is the Fourier Transform of y(s);
F is the Fourier Transform;
NOTE  The Fast Fourier Transform algorithm is employed here for discrete signals.
−1
F is the inverse of the Fourier Transform.
An example of the simple frequency domain method in MATLAB language is given in Clause A.2.
7.3 Verification of colouring process
7.3.1 Introduction
The verification of a colouring process can be done in two ways:
— using test signals;
— through verifications on recorded track geometry data.
The signals used in these processes need to cover the full wavelength range of interest.
The process can be described in Figure 10, the colouring and recolouring process is then either done by the
measurement system or by signal processing. The detail of the recolouring process is given in Clause 8.
Key
1 test signal
2 colouring process with A base
3 colouring process with B base
4 recolouring process from B base to A base
5 compute the error due to colouring and recolouring process
Figure 10 — Verification of colouring process with test signal
For both the test signal and the track measurement signal, the error signal needs to be neglectable with
respect to the maintenance values on the corresponding network.

7.3.2 Verification with test signals
The offline verification as shown in Figure 10 is possible if a test signal containing undistorted (i.e. not
affected by transfer functions of chord measurement systems) track geometry with all relevant wavelengths
is available.
The test signal can be a simulated signal, a real measurement recorded by a measurement system, or a well-
chosen test signal.
The comparison of signals at point 5 can be done in the space domain and in the frequency domain by
computing the transfer function and coherence function between the output signals of points 2 and 4.
For a pure mathematical signal, a random signal or a deterministic signal can be used, for example. Those
signals need to cover the relevant wavelength range completely.
It is also possible to use a test signal with its main energy close to the low and high gains of both chord
measurement systems A and B (see Table 2). For example, if A is a [4,5/5,5] chord measurement system and
B is a [6,5/3,5] chord system, the gain and phase of the signal at the maximum and minimum frequency of
system "A plus B" can be checked by inserting a pure sinusoidal signal with the wavelength given in Table 2
as input. Corresponding transfer functions are plotted in Figure 11.
Table 2 — Example of gain and phase of 4,5 m/5,5 m and 6,5 m/3,5 m chord measurement system
Wavelength System A System B Reason
Expected gain Expected phase Expected gain Expected phase
m rad rad
5 0,618 1,885 1,618 −2,513 max for B and min for A
3,333 1,782 −2,670 0,313 1,728 max for A and min for B
2,5 1,176 2,199 1,902 2,827 max for B and min for A
2 1,414 −2,356 1,414 −2,356 max for A and B
1,667 1,618 2,513 0,618 1,885 max for A and min for B
1,43 0,908 −2,042 1,975 2,985 max for B and min for A
1 2,00 π 2,00 π max for A and B

Key
m magnitude of the transfer function
w wavelength (m)
1 transfer function for 4,5 m/5,5 m – system A
2 transfer function for 6,5 m/3,5 m – system B
Figure 11 — Transfer functions of 4,5 m/5,5 m and 6,5 m/3,5 m chord measurement system
An example of this treatment using both a random signal and a track signal in MATLAB language is given in
Annex B.
7.3.3 Verification with recorded track geometry data
As the colouring process is a linear filter, the process described in ISO 23054-1:2022, E.3.3 can be applied here.
The principle is to compare the difference of the alignment of both rails with track gauge signal, and the
difference of both rails' longitudinal level with track cross level.
The colouring process can be applied before or after comparing the difference of both rails' alignments or
longitudinal level (Figure 12). The same colouring process has to be applied to track gauge and cross level,
before comparison.
Key
1 alignment or longitudinal level of the right rail
2 alignment or longitudinal level of the left rail
3 coloured difference of 1 and 2
4 track gauge or cross level
5 coloured track gauge or cross level
6 result of the comparison of both signals
Figure 12 — Verification of colouring process with recorded data
8 Recolouring process
8.1 Definition of recolouring process
As shown in Figure 1, recolouring is a method used to obtain the chord-based method evaluation signal
of different chord lengths or divisions from the measurement data of the track geometry by the chord
measurement system of a different chord length. For example, it is used to convert the measured data from
the track geometry car with the asymmetrical chord method into the evaluation waveform of the 10 m
symmetrical chord method (Figure 13). Alternatively, it can also be applied when converting the evaluation
waveform of the 10 m symmetrical chord method to the evaluation waveform of the 40 m symmetrical
chord method.
The signal coming from the measurement system is to be decoloured as an inertial signal according to the
properties of the measurement system and then coloured again according to the new measurement system.

Key
1 chord-based signal of the measurement system with [a , b ] chords
m m
2 decolouring of the measurement signal
3 colouring with chord [a , b ]
e e
4 evaluation signal with chord [a , b ]
e e
Figure 13 — Conversion of chord measurement data into a different chord-based system
[1][4]
However, this operation is mathematically complex since it requires the computation of the inverse of
a Fourier transform with zero or small values (for some chord division). This can be done using the Wiener
formalism, but it requires an evaluation of the noise over signal ratio. This evaluation can be done by
comparing the difference of the alignment of both rails with the gauge, and the difference of levelling of both
rails with the cant.
The following formulae provide a description of how Wiener formalism is utilized in chord-based methods.
The Laplace transform function of the measurement system is shown in Formulae (12) and (13):
b a
 
m m
Hp() = expe()−ap + xp()bp −1 (12)
m  m m 
ab+ ab+
 
mm mm
2π
px=+ jxω=+ j (13)
λ
Knowing the noise over curvature signal of the measurement system, for example in the form of their
Sp
()
nn
respective power spectral density: , the Wiener inverse H of H can be expressed as
w m
Sp()
mm
Formula (14):
Hp()
w m
Hp()= (14)
m
Sp
Hp ()
()
2 nn
m
Hp +
()
m
Sp()
mm
If only the noise over signal ratio is known, R , the previous Formula (14) can be approximated by
n/s
Formula (15).
Hp
1 ()
w m
Hp = (15)
()
m
Hp()
m
Hp +R
()
mn/s
Hence the full conversion process between the measured signal and the evaluation signal is summarized in
Formula (16).
b a
 
e e
Hp()=Hp()*Hp()= expe−ap + xp bp −1 *Hp() (16)
() ()
em ew  e e  w
ab+ ab+
 
ee ee
Figure 14 shows as an example, the recoloured waveform of the 10 m symmetrical chord method and 40 m
chord symmetrical chord delivered from the waveform of the 2,5 m/17,5 m asymmetrical chord method on
a short track section.
Key
1 2,5 m/17,5 m asymmetrical chord waveform
2 recoloured 10 m symmetrical chord waveform
3 recoloured 40 m symmetrical chord waveform
d distance (m)
y amplitude (mm)
a
Figure 14 — Example of distortion due to recolouring process
8.2 Recolouring method
The recolouring method is essentially the same method as that of the colouring method in terms of signal
processing. The method described in 7.2 can be applied here.
However, the existence of the IIR decolouring filter and the stability of the FIR decolouring filter are not
mathematically guaranteed (see Clause B.5 for further discussion). Decolouring will essentially be made in
the Laplace or Fourier domain. Therefore, the expression of an exact recolouring filter as an FIR or an IIR is
not given.
8.3 Verification of recolouring process
The verification of the recolouring process is closely related to the verification of the colouring process in
terms of signal processing. The process described in 7.3 can be applied here.
9 Elimination of influence of track geometry layout
The measured signal by the chord-based method includes the offset due to track layout components,
especially for track sections with small curve radii. Additionally, the longer the chord to be evaluated, the
larger the offset.
Therefore, to evaluate the variation of track geometry, eliminating the offset due to design components is
necessary via methods such as FIR low pass filters (e.g. moving average).
In some cases, such as when evaluating running safety, a raw signal that includes the offset due to design
components is suitable. Elimination of offset can be performed if necessary.

10 Usage of restored waveform
In simulations of vehicle dynamics, it is necessary to use the decoloured track geometry (the restored
waveform).
A method using W , W and W waveforms as shown in ISO 23054-1 also exists. This method produces
1 2 3
signals narrowed down to a specific wavelength band.
In Japan, signals that include a wide wavelength band, and all W to W waveforms, can be used. This signal
1 3
is called the "restored signal".
For e
...