ISO 16063-43:2015

DRAFT INTERNATIONAL STANDARD
ISO/DIS 16063-43
ISO/TC 108/SC 3 Secretariat: DS
Voting begins on: Voting terminates on:
2014-11-12 2015-02-12
Methods for the calibration of vibration and shock
transducers —
Part 43:
Calibration of accelerometers by model-based parameter
identification
Méthodes pour l’étalonnage des transducteurs de vibrations et de chocs —
Partie 43: Étalonnage des accéléromètres par identification des paramètres à base de modèle
ICS: 17.160
THIS DOCUMENT IS A DRAFT CIRCULATED
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NATIONAL REGULATIONS.
ISO/DIS 16063-43:2014(E)
RECIPIENTS OF THIS DRAFT ARE INVITED
TO SUBMIT, WITH THEIR COMMENTS,
NOTIFICATION OF ANY RELEVANT PATENT
RIGHTS OF WHICH THEY ARE AWARE AND TO
©
PROVIDE SUPPORTING DOCUMENTATION. ISO 2014

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ISO/DIS 16063-43:2014(E)

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ISO/DIS 16063-43
Contents Page
Foreword . iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 2
4 Consideration of typical frequency response and transient excitation . 2
5 General approach . 3
6 Linear Mass-Spring-Damper Model . 3
6.1 Model . 3
6.2 Identification by sinusoidal calibration data . 4
6.2.1 Parameter identification . 4
6.2.2 Uncertainties of model parameters by analytic propagation . 6
6.3 Identification by shock calibration data in the frequency domain . 7
6.3.1 Identification of the model parameters . 7
6.3.2 Uncertainties of model parameters by analytical propagation. 10
7 Practical considerations . 11
7.1 The influence of the measurement chain . 11
7.2 Synchronicity of the measurement channels . 11
7.3 Properties of the source data used for the identification . 12
7.4 Empirical test of model and parameter validity. 12
7.4.1 Sinusoidal calibration data . 12
7.4.2 Shock calibration data . 12
7.5 Statistical test of model validity . 12
7.5.1 General . 12
7.5.2 Statistical test for sinsoidal data . 13
7.5.3 Statistical test for shock data and the frequency domain evaluation . 13
8 Reporting of results . 13
8.1 Common considerations on the reporting . 13
8.2 Results and conditions to be reported . 13
Bibliography . 16

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ISO/DIS 16063-43
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International
Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
ISO 16063-43 was prepared by Technical Committee ISO/TC 108, Mechanical vibration, shock and condition
monitoring, Subcommittee SC 3, Use and calibration of vibration and shock measuring instruments.
ISO 16063 consists of the following parts, under the general title Methods for the calibration of vibration and
shock transducers:
 Part 1: Basic concepts
 Part 11: Primary vibration calibration by laser interferometry
 Part 12: Primary vibration calibration by the reciprocity method
 Part 13: Primary shock calibration using laser interferometry
 Part 15: Primary angular vibration calibration by laser interferometry
 Part 16: Calibration by Earth's gravitation
 Part 21: Vibration calibration by comparison with a reference transducer
 Part 22: Shock calibration by comparison with a reference transducer
 Part 31: Testing of transverse vibration sensitivity
 Part 41: Calibration of laser vibrometers
 Part 42: Calibration of seismometers with high accuracy using acceleration of gravity
The following parts are under preparation:
 Part 32: Resonance testing – Testing the frequency and the phase response of accelerometer by means
of shock excitation
 Part 33: Testing of magnetic field sensitivity
 Part 43: Calibration of accelerometers by model-based parameter identification
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ISO/DIS 16063-43
Introduction
The standard series of ISO 16063 describes in several of its parts (ISO 16063-1, ISO 16063-11, ISO 16063-13,
ISO 16063-21 and ISO 16063-22) the devices and procedures to be used for calibration of vibration sensors.
The approaches taken can be divided in two classes. One for the use of stationary signals, namely sinusoidal
or multi-sinus excitation, the other for transient signals, namely shock excitation. While the first provides the
lowest uncertainties due to the intrinsic, periodic repeatability the later is aiming at the high intensity range where
periodic excitation is usually not feasible due to power constraints of the calibration systems.
The result of the first class is given in terms of a complex transfer sensitivity in the frequency domain and is
hence not directly applicable to transient time-domain application.
The results of the latter class are given as a single value, the peak ratio, in the time domain which neglects
(knowingly) the frequency dependent dynamic response of the transducer to transient input signals with spectral
components in the resonance area of the transducer's response. As a consequence of this “peak ratio
characterisation”, the calibration result might exhibit a strong dependence on the shape of the transient input
signal applied for the calibration and therefore from the calibration device.
This has two serious consequences:
1) The calibration with shock excitation according to ISO 16063-13 or ISO 16063-22 is of limited use as
far as the dissemination of units is concerned. That is, the shock sensitivities Ssh determined by
calibrations on device in a primary laboratory might not be applicable to the customer's device in the
secondary calibration lab, simply due to a different signal shape and thus spectral constitution of the
secondary device's shock excitation signal.
2) A comparison of calibration results from different calibration facilities with respect to consistency of the
estimated measurement uncertainties, e.g. for validation purposes in an accreditation process, is not
feasible if the facilities apply input-signals of differing spectral composition.
The approach taken here is a mathematical model description of the accelerometer as a dynamic system with
mechanical input and electrical output, where the latter is assumed to be proportional to an intrinsic mechanical
quantity (e.g. deformation). The estimates of the parameters of that model and the associated uncertainties are
then determined on the base of calibration data achieved with the established methods (ISO 16063-11,
ISO 16063-13, ISO 16063-21 and ISO 16063-22). The complete model with quantified parameters and their
respective uncertainties can subsequently be used to either calculate the time-domain response of the sensor
to arbitrary transient signals (including time dependent uncertainties) or as a starting point for a process to
estimate the unknown transient input of the sensor from its measured time-dependent output signal (ISO 16063-
11 or ISO 16063-13).
As a side effect, the method usually provides an estimate of a continued frequency-domain transfer sensitivity
of the model, too.
In short the methods and procedures prescribed in this document enable the user to:
 calibrate vibration transducers for precise measurements of transient input,
 perform comparison measurements for validation using transient excitation,
 predict transient input signals and its time dependent measurement uncertainty,
 compensate effects of the frequency dependent response of vibration transducers (in real time) and thus
expand the applicable bandwidth of the transducer.


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DRAFT INTERNATIONAL STANDARD ISO/DIS 16063-43

Methods for the calibration of vibration and shock
transducers — Part 43: Calibration of accelerometers by model-
based parameter identification
1 Scope
This International Standard prescribes terms and methods on the estimation of parameters used in
mathematical models describing the input-output characteristic of vibration transducers together with the
respective parameter uncertainties. The described methods estimate the parameters on the basis of calibration
data collected with standard calibration procedures according to established standards ISO 16063-1,
ISO 16063-11, ISO 16063-13, ISO 16063-21 and ISO 16063-22. The specification is provided as an extension
of the existing procedures and definitions in those standards. The uncertainty estimation described conforms to
the methods established by ISO/IEC Guide 98-3 and Supplement 1.
The new characterisation described in this document is intended to improve the quality of calibrations and
measurement applications with broadband/transient input, like shock. It provides the means of a
characterisation of the vibration transducer's response to a transient input and therefore provides a basis for the
accurate measurement of transient vibrational signals with the prediction of an input from an acquired output
signal. The calibration data for accelerometers used in the aforementioned field of applications should
additionally be evaluated and documented according to the methods described below in order to provide
measurement capabilities and uncertainties beyond the limits drawn by the single value characterisation given
by ISO 16063-13 and ISO 16063-22.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are indispensable
for its application. For dated references, only the edition cited applies. For undated references, the latest edition
of the referenced document (including any amendments) applies.
ISO 2041, Mechanical vibration, shock and condition monitoring — Vocabulary
ISO 16063-1, Methods for the calibration of vibration and shock transducers – Part 1: Basic concepts
ISO 16063-11, Methods for the calibration of vibration and shock transducers – Part 11: Primary vibration
calibration by laser interferometry
ISO 16063-13, Methods for the calibration of vibration and shock transducers – Part 13: Primary shock
calibration using laser interferometry
ISO 16063-21, Methods for the calibration of vibration and shock transducers – Part 21: Vibration calibration by
comparison to a reference transducer
ISO 16063-22, Methods for the calibration of vibration and shock transducers - Part 22: Shock calibration by
comparison to a reference transducer
ISO/IEC Guide 98-3, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
measurement (GUM:1995)
ISO/IEC Guide 98-3:2008/Suppl 1/2008, Uncertainty of measurement — Part 3: Guide to the expression of
uncertainty in measurement (GUM:1995) – Supplement 1: Propagation of distributions using a Monte Carlo
method
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ISO/DIS 16063-43
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 2041 apply.
4 Consideration of typical frequency response and transient excitation
A typical acceleration transducer has a complex frequency response. This is usually given in terms of magnitude
and phase with a shape as it is depicted in Figure 1. The magnitude is given in arbitrary units (a.u.).
This response function is subsequently sampled with lowest uncertainties by a calibration method according to
ISO 16063-11 or ISO 16063-21 making use of periodic excitation.
In applications with transient input-signals such a sensor is then exposed to broadband excitation in terms of
the frequency domain. The response in this case cannot be calculated with the help of a single (complex) value
like the transfer sensitivity. Rather, the response can be considered to be a sensitivity that is weighted by those
components in the frequency response which are excited by the spectral contents of the input signal.


Figure 1 — Complex frequency response of a typical accelerometer in terms of magnitude of
sensitivity (blue) and phase delay (green) over the normalized frequency

Figure 2 gives a pictorial representation of three examples of possible shock excitation signals and their
respective spectra as compared to the frequency response of a typical sensor. It shows the projection of the
centre of mass of the magnitude of the spectral density curve onto the sensitivity curve of a typical accelerometer.
This demonstrates, that a single value characterisation of a transducer by shock calibration cannot sufficiently
describe the dynamic behaviour.
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ISO/DIS 16063-43




Figure 2 — Time domain representation (left) of three different shock signals (long monopole (red),
medium dipole (green), short dipole (blue)) with the respective magnitude spectra (middle) with the
projection of the spectral centre point onto the sensitivity curve of a typical accelero¬meter
response, and the corresponding shock sensitivity (peak ratio) of a typical accelerometer (right)

5 General approach
The general idea behind “model based parameter identification” is to describe the input/output behavior of a
transducer type of certain design and construction with the help of a dynamic mathematical model. The detailed
1
properties of an individual transducer are represented in that model by a set of parameters . Associated with
the set of estimates of the model parameters is a respective set of uncertainties. The aim of the calibration is to
provide measurement results which allow for the mathematical estimation of this parameter set and the
evaluation of corresponding uncertainties.
This general approach is not new but already well established in science and engineering under the term
“identification of dynamic systems”. However, in the field of transducer calibration special emphasis has to be
put on the validation of the applicability of the methods used and on the reliable calculation of uncertainties and
respective coverage intervals.
NOTE In the subsequent text the procedure of model based parameter identification and further considerations are
presented for a linear mass-spring-damper model of a seismic pick-up. However, this is only one example. The same
approach can be used for more complicated mathematical models as long as they can be described as linear time-invariant
(LTI) systems.
6 Linear Mass-Spring-Damper Model
6.1 Model
According to the investigation in [8,10 and 13] some accelerometers can be described by a simple linear mass-
spring-damper model in their specified working range. That means they follow the general equation of motion
of the form (c.f. [2])
2
𝑥¨+2𝛿𝜔 𝑥˙+𝜔 𝑥=𝜌𝑎(𝑡) (1)
0 0
with δ being the damping coefficient, ω the circular resonant frequency of the system and ρ the electro
0
mechanical conversion factor. This model describes the dynamic output 𝑥(𝑡) (e.g. charge or voltage) as a
function of the acceleration input 𝑎(𝑡)

1 The parameter sets may include functions of variables to cover temperature sensitivity or mass loading effects.
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ISO/DIS 16063-43
For such a linear system the transfer function 𝐻(iω) in the frequency domain is independent of the acceleration
amplitude and is given by:
ρ
iϕ(ω)
H(iω)= =S(ω)⋅e . (2)
2
2
ω +2δω iω+(iω)
0
0
The inverse of this transfer function is:
−1 −1 2 2 −1 −iϕ(ω)
G(iω)=H(iω) =ρ (ω +2iωδω −ω )=S (ω)⋅e (3)
0 0
with S(ω) representing the magnitude and ϕ(ω) representing the phase of the response.
6.2 Identification by sinusoidal calibration data
6.2.1 Parameter identification
Starting from calibration measurements with sinusoidal excitation according to e.g. ISO 16062-11 or
ISO 16062-21 one can directly determine the frequency response 𝐻(𝑖𝜔) as described by Equation (2) taking
2
into account the well known frequency response of any conditioning amplifier .
Substituting a parameter vector
2
𝜔 2𝛿𝜔 1
0
𝑇 0
𝜇 =(𝜇 ,𝜇 ,𝜇 )=( , , ) (4)
1 2 3
𝜌 𝜌 𝜌
Equation (3) transforms to
1
2 𝑇
𝐺(𝑖𝜔)= =𝜇 +𝑖𝜇 𝜔−𝜇 𝜔 =𝑔 (𝜔)⋅𝜇 (5)
1 2 3
𝐻(𝑖𝜔)
𝑇 2
with 𝑔 (𝜔)=(1,𝑖𝜔,−𝜔 ).

According to this relation the parameter vector 𝜇 can be estimated by weighted linear least squares, where the
weights are chosen according to the uncertainties known from the calibration procedures according to
ISO 16062-11 or ISO 16062-21 as follows.
Let 𝑆 =𝑆(𝜔 ), 𝜙 =𝜙(𝜔 ) denote the magnitude and the phase of the frequency response from calibration
𝑚 𝑚 𝑚 𝑚
measurements with associated standard uncertainties 𝑢(𝑆 ), 𝑢(𝜙 ) at the frequencies 𝜔 , 𝑚=1,2,.,𝐿. Then
𝑚 𝑚 𝑚
−1 −𝑖𝜙 −1 −𝑖𝜙
the real part 𝑅(𝑆 ⋅𝑒 ) and imaginary part 𝐽(𝑆 ⋅𝑒 ) are given by:
−1 −𝑖𝜙 −1
    𝑅(𝑆,𝜙)=𝑅(𝑆 ⋅𝑒 )=𝑆 𝑐𝑜𝑠(𝜙) ,
−1 −𝑖𝜙 −1
𝐽(𝑆,𝜙)=Im(𝑆 ⋅𝑒 )=−𝑆 𝑠𝑖𝑛(𝜙) (6)
This is in principle a non-linear transform which should be adequately handled for uncertainty calculations by
ISO/IEC Guide 98-3, Supplement 1. However, given the case that the uncertainties of measurement are small
enough the direct propagation of uncertainties can be calculated according to ISO/IEC Guide 98-3 as:
2 2
𝑢 (𝑆 ) 𝑢 (𝜙 )
𝑚 𝑚
2 2 2
𝑢 (𝑅 ) = 𝑐𝑜𝑠 (𝜙 )+ 𝑠𝑖𝑛 (𝜙 )
𝑚 𝑚 𝑚
4 2
𝑆 𝑆
𝑚 𝑚
2 2
𝑢 (𝑆 ) 𝑢 (𝜙 )
𝑚 𝑚
2 2 2
𝑢 (𝐽 ) = 𝑠𝑖𝑛 (𝜙 )+ 𝑐𝑜𝑠 (𝜙 )
(7)
𝑚 𝑚 𝑚
4 2
𝑆 𝑆
𝑚 𝑚
2 2
−𝑢 (𝑆 ) 𝑢 (𝜙 )
𝑚 𝑚
𝑢(𝑅 ,𝐽 ) = 𝑠𝑖𝑛(𝜙 )𝑐𝑜𝑠(𝜙 )+ 𝑠𝑖𝑛(𝜙 )𝑐𝑜𝑠(𝜙 )
𝑚 𝑚 𝑚 𝑚 𝑚 𝑚
4 2
𝑆 𝑆
𝑚 𝑚

2
The model assumes that any additional response function of a measuring amplifier is eliminated prior to the identification
process, which is usually the case.
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ISO/DIS 16063-43
with 𝑅 =𝑅(𝑆 ,𝜙 ) and 𝐽 =𝐽(𝑆 ,𝜙 )
𝑚 𝑚 𝑚 𝑚 𝑚 𝑚
Then let
𝑇
𝑦 =(𝑅(𝑆 𝜙 ),…,𝑅(𝑆 ,𝜙 ),𝐽(𝑆 𝜙 ),…,𝐽(𝑆 ,𝜙 )) (8)
1, 1 𝐿 𝐿 1, 1 𝐿 𝐿
be the transformed vector of the measurands. With the assumption that 𝑆 and 𝜙 are uncorrelated measurands
its 2L×2L covariance matrix 𝑉 becomes:
𝑦
2
𝑢 (𝑅 ) 0 𝑢(𝑅 𝐽) 0
1 1, 1
⋱ ⋱

2
0 𝑢 (𝑅 ) 𝑢(𝑅 ,𝐽 )
𝐿 𝐿 𝐿
𝑉 = (9)
𝑦
2
𝑢(𝑅 𝐽) 𝑢 (𝐽) 0
1, 1 1

⋱ ⋱
2
( 0 𝑢(𝑅 ,𝐽 ) 0 𝑢 (𝐽 ) )
𝐿 𝐿 𝐿
𝑇
With 𝐻 being the 2L times 3 matrix of the real and imaginary parts of 𝑔 (𝜔)
2
1 0 −𝜔
1
2
1 0 −𝜔
2

⋮

2

1 0 −𝜔
𝐿
𝐷= (10)

0 𝜔 0
1

0 𝜔 0
2
⋮
(0 𝜔 0 )
𝐿
the weighted least square estimate of the parameters can be calculated according to
−1
𝑇 −1 𝑇 −1
𝜇ˆ=(𝐷 𝑉 𝐷) 𝐷 𝑉 𝑦 (11)
𝑦 𝑦
The uncertainties associated with the estimated parameter set 𝜇ˆ, i.e., the covariance matrix is given by
𝑇 −1 −1
𝑉 =(𝐷 𝑉 𝐷) (12)
𝜇ˆ 𝑦
The original model parameters can subsequently be calculated by transforming Equation (4) as
−1
𝜌 = 𝜇
3
𝜇
1
𝜔 =
√
0
(13)
𝜇
3
𝜇
2
𝛿 =
√𝜇 ⋅𝜇
1 3
2
Sometimes it is more convenient to write (2) in terms of a 𝑆 =𝜌/𝜔 instead of 𝜌 where 𝑆 describes the
0 0 0
sensitivity for low frequencies. The corresponding parameter equation is
1
𝑆 = . (14)
0
𝜇
1
Since the inverse transform from Equation (4) to Equation (13) is non-linear the uncertainties associated with
the model parameter set should be adequately handled for uncertainty calculations by e.g. Monte Carlo methods
as described in [1].
Figure 3 gives a flow chart representation of the whole analysis process.
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ISO/DIS 16063-43


Figure 3 — Flowchart of the process of parameter identification upon sinusoidal calibration data
6.2.2 Uncertainties of model parameters by analytic propagation
In cases where the total expanded relative uncertainty of measurement of the magnitude 𝑆 is less than 1 %
𝑚
and the total uncertainty of measurement of the phase 𝜙 is less than 2° a conventional propagation of
𝑚
uncertainty is feasible, although Equation (13) states a strongly nonlinear relationship.

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With the transformation matrix
∂𝜌 ∂𝜌 ∂𝜌
∂𝜇 ∂𝜇 ∂𝜇
1 2 3

∂𝜔 ∂𝜔 ∂𝜔
0 0 0
𝐴 = (15)
𝜇

∂𝜇 ∂𝜇 ∂𝜇
1 2 3
∂𝛿 ∂𝛿 ∂𝛿
(∂𝜇 ∂𝜇 ∂𝜇 )
1 2 3
The Covariance matrix of the model parameters 𝑉 can be calculated from the covariance matrix 𝑉 by
𝜌,𝜔 𝛿 𝜇ˆ
0,
𝑇
𝑉 =𝐴 𝑉𝐴 (16)
𝜌,𝜔 𝛿 𝜇 𝜇ˆ 𝜇
0,
where the square roots of the diagonal elements of 𝑉 state the uncertainties of the model parameters.
𝜌,𝜔 𝛿
0,
The uncertainty for 𝑆 can be calculated accordingly by substituting 𝑆 for 𝜌 in Equation (15).
0 0
NOTE The non linear relationship in Equation (13) and Equation (14) requires an appropriate handling of the
uncertainty propagation according to ISO/IEC Guide 98-3/Supplement 1 for the general case. Only in the case of reduced
input uncertainties the linearisation described in 7.2.2 is applicable. For the given model a comprehensive description of the
general case is given in [1].

6.3 Identification by shock calibration data in the frequency domain
6.3.1 Identification of the model parameters
Starting from calibration measurements with shock excitation according to e.g. ISO 16063-13 or ISO 16063-22
it is possible to estimate the model parameters using a special preprocessing step with subsequent identification
similar to the procedure described in the previous clause.
For the substitution a discretisation of the continuous time Equation (1) is necessary. For that purpose the
classical s-transform is used, which leads in case of Equation (1) to the transformed equation:
2 2
(𝑠 +2𝛿𝜔 𝑠+𝜔 )𝑋(𝑠)=𝜌𝐴(𝑠) (17)
0 0
The discretization follows e.g. by a bilinear mapping of the s-plane to the z-plane of the kind:
−1
2 1−𝑧
𝑠→ ⋅ (18)
−1
𝑇 1+𝑧
where T is the sampling interval.

By substituting Equation (18) into Equation (17), taking the sampled time series 𝑥 and 𝑎 for the respective
𝑖 𝑖
−1 −1
variables and applying the back shift operator 𝑧 , (𝑧 ∙𝑥 = 𝑥 ) of the z-transform properly, one arrives at
𝑖 𝑖−1
3
the discretised version of the model equation (c.f. [3]) of the form:
𝑥 =−𝑐 𝑥 −𝑐 𝑥 +𝑏(𝑎 +2𝑎 +𝑎 ) (19)
𝑘 1 𝑘−1 2 𝑘−2 𝑘 𝑘−1 𝑘−2
The parameters of Equation (19) are related to the continuous model parameters according to

3
The error introduced by the discretization with respect to the sample rate needs some further investigation
and consideration in the uncertainty budget. At the time of writing no publications available on this topic.
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ISO/DIS 16063-43
2
𝜌𝑇
𝑏 =
4𝛬
2
2
𝜔 𝑇 −4
0
𝑐 =
1
2𝛬
2 2
4−4𝛿𝜔 𝑇+𝜔 𝑇 . (20)
0
0
𝑐 =
2
4𝛬
with
2 2
𝜔 𝑇
0
𝛬 = 1+𝛿𝜔 𝑇+
0
4
It is clear from this derivation that the parameters of the discretised model are dependent upon the sample rate
−1
𝑇 and are therefore closely related to the calibration set-up. The sample rate used for the measurement should
be at least five times greater than the frequency at which the first significant resonance of the sensor under
calibration occurs. In order to avoid additional significant uncertainty components due to lack of resolution over
frequency regions in which resonances occur, a better sampling frequency would be a factor of 10 or more
greater than the frequency at which the first significant resonance of the sensor under calibration occurs.
The introduced discrete time model Equation (19) has a (periodic) frequency response of the form:
−𝑗𝛺 −𝑗2𝛺
𝑏(1+2𝑒 +𝑒 )
𝑗𝛺
𝐻(𝑒 )= (21)
−𝑗𝛺 −𝑗2𝛺
1+𝑐 𝑒 +𝑐 𝑒
1 2
where 𝛺=𝜔/𝑓 =𝜔𝑇 is the radian frequency normalized to the sample rate.
𝑠
𝑗𝛺
Just like Equation (5) in 7.2 the inverse of this frequency response 𝐺(𝑒 ) is linear in the parameters. After some
obvious substitution:
−𝑗𝛺 −𝑗2𝛺 −𝑗𝛺 −𝑗2𝛺
1+𝑐 𝑒 +𝑐 𝑒 𝜈 +𝜈 𝑒 +𝜈 𝑒
1 2 1 2 3
𝑗𝛺 −1 𝑗𝛺
𝐺(𝑒 )=𝐻 (𝑒 )= = (22)
−𝑗𝛺 −𝑗2𝛺 −𝑗𝛺 −𝑗2𝛺
𝑏(1+2𝑒 +𝑒 ) (1+2𝑒 +𝑒 )
with the substitute vector
𝑇
[ ]
𝜈 = 𝑣 ,𝑣 ,𝑣 =[1/𝑏,𝑐 /𝑏,𝑐 /𝑏]. (23)
1 2 3 1 2
With this inverse frequency response the general approach taken already in 7.2 can be followed. For the sake
of completeness this will be worked out in more detail in the following.
Let 𝑋(𝑛) and 𝐴(𝑛) be the components of the discrete Fourier transform (DFT) of the sampled time series 𝑥
𝑘
and 𝑎 respectively with 𝑛=0, 1, …, 𝑁−1. Here, the influence of a conditioning amplifier can be eliminated
𝑘
1 𝑛
by multiplying A(n)by the measured complex frequency response of the amplifier for the frequency  in order
2𝑇𝑁
to compensate the response later in Equation (24). In cases where AC-coupled conditioning amplifiers are used,
the terms for n = 0 should be omitted, because they describe the DC-component of the signals which vanishes
for 𝑋 . Equation (22) implies the relation
0
𝐴(𝑛)
𝑇
𝐺 = =𝑓 𝜈 (24)
...