The European Centre for
Mathematics and Statistics in Metrology

Analysis of dynamic measurements


Dynamic measurements can be found in many areas of metrology and industry, such as, for instance, in applications where mechanical quantities, electrical pulses or temperature curves are measured.
A quantity is called dynamic when its value at one time instant depends on its values at previous time instants

That is, in contrast to static measurements where a single value or a (small) set of values is measured, dynamic measurements consider continuous functions of time. Since the analysis of dynamic measurements requires different approaches than the analysis of static measurements this part of metrology is often called "Dynamic Metrology". The mathematical modeling of dynamic measurements typically utilizes methodologies and concepts from digital signal processing. In the language of metrology a signal denotes a dynamic quantity, and a system a measurement device whose input and/or output are signals. The output signal of a system is thus the indication value of the measurement device for a corresponding input signal.

In mathematical terms the signals are continuous time dependent functions $x(t)$ and $y(t)$. In most metrological applications the measurement system can be considered time-invariant and linear with respect to its inputs: $$ \mathcal{H}\left( a_1 x_1(t) + a_2 x_2(t) \right) = a_1\mathcal{H}(x_1(t)) + a_2 \mathcal{H}(x_2(t))$$ Such systems are called linear time-invariant (LTI) and are fully represented by their impulse response function $h(t)$, equivalently by their transfer function $H(s)$ or frequency response function $H(f)$. The relation between input and output signal is then given mathematically as a convolution $$x(t) = (y\ast h)(t) = \int_{-\infty}^{\infty} y(s)h(s-t)ds $$.

A characteristic property of a dynamic measurement is that the output signal is not proportional to the input signal owing to dynamic effects caused by the measurement system. For instance, accelerometers typically show a resonance behavior. For a measured acceleration with a certain frequency content the output signal of the accelerometer then shows a significant "ringing" [Elster et al. 2008].

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Typical dynamic measurement with time dependent errors in the output signal caused by the dynamic behavior of the measurement system.
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Amplitude of the frequency response of the measurement system, the compensation filter and the compensated measurement system.
The aim in the analysis of dynamic measurements is the compensation for time dependent errors, such as, ringing, phase deviations and others. In contrast to static measurements, this cannot be accomplished by scaling and shifting the output signal. Instead, for linear time-invariant systems (LTI) a so called deconvolution has to be carried out [Riad 1986]. This allows the compensation of dynamic effects and thereby, in principle, the reconstruction of the actual input signal.
The analysis of dynamic measurements in the case of linear time-invariant (LTI) systems can be carried out by application of a suitable digital filter. The design of such a filter is based on the available knowledge about the measurement system and aims at a compensation of its undesired dynamic behaviour [Eichstädt et al. 2010]. Such an approach is typically applied when an appropriate physical model of the measurement system is available. For instance, in the dynamic measurement of mechanical quantities the sensor is usually modelled as a combination of one or several mass-damper-spring elements and the involved mass bodies. The resulting mathematical model can then be translated into a dynamic system model of the assumed LTI system, see, e.g., [Schlegel et al. 2012, Kobusch et al. 2015, Klaus et al., 2015].
As illustrated in the example in Fig. 2, the frequency response of the compensation filter is the reciprocal of the system's frequency response up to a certain frequency. Thus, the prerequisite for the design of a compensation filter is a dynamic calibration of the measurement device in a suitable frequency range.
The same holds true in the case of deconvolution in the Fourier domain; where the measured time domain system output signal is transformed using the DFT and deconvolution is carried out by division in the frequency domain. This approach is taken typically when there is no simple parametric model available that represents the dynamic behaviour of the measurement system in the desired accuracy. Examples are calibration of sampling oscilloscopes [Dienstfrey et al. 2006, Hale et al. 2012, Füser et al. 2012] or hydrophones [Wilkens et al. 2004, Wear et al. 2015].
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Typical workflow in the analysis of dynamic measurements. The response of the system to the continuous-time values of the measurand are sampled by an analogue-to-digital converter (A/D) and a discrete-time estimate of the values of the measurand are calculated.
The literal meaning of "dynamic" relates solely to time varying quantities. However, from a mathematical perspective the definition of a dynamic measurement can be extended to other independent quantities as time. This includes quantities whose value depend on frequency, spatial coordinates, wavelength, etc. The extended definition is reasonable, because the mathematical treatment of such measurements does not depend on the physical interpretation of the independent quantity.
A quantity is called dynamic if its values depend on another, independent, quantity. A measurement is dynamic if at least one of the involved quantities is dynamic.
The extended definition of a dynamic measurement contains a wide spectrum of metrological applications. Typical examples are measurements of mechanical quantities, high-speed electronics, medical ultra-sound, spectral characterisation of radiation sources. The applications range from single sensor measurements up to large sensor networks. However, the approach to estimating the dynamic measurand may vary with the physical interpretation. For instance, deconvolution for bandwidth correction in spectrometry and radiometry requires one to restrict the solution to contain only non-negative values due to physical reasons, see, e.g., [Eichstädt et al. 2013].
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Difference between the (time shifted) output signal and the above input signal with and without application of the compensation filter

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Actual and measured spectral power distribution of a light source



The analysis of dynamic measurements leads to a number of scientific challenges for metrologists. MATHMET members are actively contributing to this area by providing mathematical and statistical methods and guidance for metrologists.

Scientific research of MATHMET members in this field contributed to projects such as EMRP IND09, EMRP IND16, EMRP ENG63 and EMPIR 14SIP08. Research topics are, for instance, methods for the statistical analysis of dynamic calibration, design of digital deconvolution filters for estimating the value of the measurand, GUM compliant evaluation of dynamic measurement uncertainty and efficient implementation of GUM Monte Carlo for the application of digital

In order to make the utilisation of the developed methods as easy as possible, MATHMET members published a number of freely available software. In addition, a best practice guide for industrial dynamic measurements is available at the website of the EMRP IND09 project.

Assigning measurement uncertainty

In principal, the value of a dynamic quantity is a continuous function of time, frequency, space, temperature or another independent quantity, respectively a tuple of such. This is not covered by the GUM framework. However, it has been shown that a consistent extension of the GUM framework to stochastic processes as model for the state-of-knowledge could be applied to this end (PhD Thesis). The consideration of stochastic processes does also cover a consistent treatment of interpolation, either by deterministic functions (e.g. calibration curves) or by stochastic processes (e.g. Gaussian process regression).

Propagation of uncertainties

In general, evaluation and propagation of measurement uncertainties for discretized dynamic quantities can be carried out by application of the GUM framework. However, there are many mathematical and practical challenges which require specific developments and research. For instance, in practice the uncertainty associated with a static quantity is determined by repeated measurements. Therefore, for univariate quantities a rather small number of measurements is sufficient. For multivariate quantities, however, the necessary number of measurements increases with the dimension of the quantity. Discretized dynamic quantities are typically very high dimensional, with a typical dynamic measurement consisting of more than thousand time instants. The evaluation of uncertainty by means of repeated measurements is thus not possible. To this end, parametric approaches have to be determined. Corresponding methods can be found in the field of time series analysis, but their application in metrology requires significant further developments.

Estimating the measurand

In most cases estimation of the measurand in dynamic measurements requires a deconvolution. However, this is a mathematically ill-posed inverse problem. That is, it requires some kind of regularization in order to obtain reasonable uncertainties. To this end, a typical approach in signal processing is the application of a suitable low-pass filter. In fact, many classical concepts of deconvolution such as Tikhonov regularization or Wiener deconvolution can be interpreted as a successive application of the reciprocal system response and a low-pass filter.  However, taking into account prior knowledge about the measurand is currently not considered in the GUM and its supplements. The type of prior knowledge can be, for instance, a parametric model or an upper bound in the frequency domain. In every case the low-pass filter causes a systematic deviation in the estimation result. For metrological applications, these systematic errors have to be considered in the uncertainty budget. However, so far no harmonized treatment of these uncertainty contributions is available.

Related journal papers

M. Kobusch, S. Eichstädt, L. Klaus, T. BrunsInvestigations for the model-based dynamic calibration of force transducers by using shock excitationACTA IMEKO2015
L. Klaus, B. Arendacká, M. Kobusch and T. BrunsDynamic torque calibration by means of model parameter identificationACTA IMEKO2015
T. Esward, C. Matthews, S. Downes, A. Knott, S. Eichstädt and C. ElsterUncertainty evaluation for traceable dynamic measurement of mechanical quantities: A case study in dynamic pressure calibrationAdvanced Mathematical & Computational Tools in Metrology and Testing IX" , Series on Advances in Mathematics for Applied Sciences vol. 84, eds. F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, K. Shirono. World Scientific New Jersey2012
S. Eichstädt and C. ElsterUncertainty evaluation for continuous-time measurementsAdvanced Mathematical and Computational Tools in Metrology and Testing IX2012
S. Eichstädt, B. Arendacká, A. Link and C. ElsterEvaluation of measurement uncertainties for time-dependent quantitiesEPJ Web of Conferences 772014
M. Musi? and M. Ahi?-Džoki? and Z. Džemi?A new approach to detection of vortices using ultrasoundFlow Measurement and Instrumentation2015
D.A. Humphreys, P.M. Harris, M. Rodriguez-Higuero, F.A. Mubarak, D. Zhao and K. OjasaloPrincipal component compression method for covariance matrices used for uncertainty propagationIEEE Transactions on Instrumentation and Measurement, vol 64(2)2014
K. A. Wear, Y. Liu, P. M. Gammell, S. Maruvada, and G. R. HarrisCorrection for Frequency-Dependent Hydrophone Response to Nonlinear Pressure Waves Using Complex Deconvolution and Rarefactional Filtering: Application With Fiber Optic Hydrophones KeithIEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL2015
A. Dienstfrey, P. D. HaleColored Noise and Regularization Parameter Selection for Waveform MetrologyInstrumentation and Measurement, IEEE Transactions on2014
P. D. Hale, A. Dienstfrey, J. Wang, D. F. Williams, A. Lewandowski, D. A. Keenan, T. S. ClementTraceable Waveform CalibrationWith a Covariance-Based Uncertainty AnalysisInstrumentation and Measurement, IEEE Transactions on2009
S. Eichstädt and C. ElsterReliable uncertainty evaluation for ODE parameter estimation - a comparisonJ. Phys. 490, 1, 0122302014
S. Eichstädt, C. Elster, I.M. Smith, T.J. EswardEvaluation of dynamic measurement uncertainty – an open-source software package to bridge theory and practice.J. Sens. Sens. Syst., 6 97-1052017
S. Eichstädt and V. WilkensGUM2DFT - A software tool for uncertainty evaluation of transient signals in the frequency domainMeas. Sci. and Technol., 27(5)2016
B. Arendacká, A. Täubner, S. Eichstädt, T. Bruns and C. Elster Linear mixed models: GUM and beyondMeas. Sci. Rev. 14, 52-612012
G. H. Nam, M. G. Cox, P. M. Harris, S. P. Robinson, G. Hayman, G. A. Beamiss, T. J. Esward and I. M. SmithA model for characterizing the frequency-dependent variation in sensitivity with temperature of underwater acoustic transducers from historical calibration dataMeas. Sci. Technol. 18, 1553-15622007
H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschke, K. Pierz and M. BielerOptoelectronic time-domain characterization of a 100 GHz sampling oscilloscopeMeas. Sci. Technol. 23, 0252012012
S. Eichstädt, N. Makarava and C. ElsterOn the evaluation of uncertainties for state estimation with the Kalman filterMeas. Sci. Technol. vol. 27(12), 125009, 20162016
Collett M, Esward T J, Harris P M, Matthews C E, Smith I MSimulating distributed measurement networks in which sensors may be faulty, noisy and interdependent: A software tool for sensor network design, data fusion and uncertainty evaluationMeasurement2013
S. Eichstädt, A. Link, T. Bruns and C. Elster On-line dynamic error compensation of accelerometers by uncertainty-optimal filteringMeasurement 43, 708-7132010
J. P. HesslingA novel method of estimating dynamic measurement errorsMeasurement Science and Technology2006
J. P. HesslingA novel method of dynamic correction in the time domainMeasurement Science and Technology2008
J. P. HesslingDynamic metrology—an approach to dynamic evaluation of linear time-invariant measurement systemsMeasurement Science and Technology2008
J. P. HesslingA novel method of evaluating dynamic measurement uncertainty utilizing digital filtersMeasurement Science and Technology, 20, nr. 5, 0551062009
J. P. HesslingDynamic calibration of uni-axial material testing machinesMechanical systems and signal processing, 22, nr. 2, 451-4662008
C. Elster, A. LinkUncertainty evaluation for dynamic measurements modelled by a linear time-invariant systemMetrologia2008
A. Link, C. ElsterUncertainty evaluation for IIR (infinite impulse response) filtering using a state-space approachMetrologia2009
C. Schlegel, G. Kieckenap, B. Glöckner, A. Buß and R. Kumme Traceable periodic force calibrationMetrologia2012
S. Eichstädt, C. Elster, T. J. Esward and J. P. HesslingDeconvolution filters for the analysis of dynamic measurements: a tutorialMetrologia 47, 522-5332010
T. Bruns, A. Link and A. TäubnerThe influence of different vibration exciter systems on high frequency primary calibration of single-ended accelerometersMetrologia 49, 27-312012
S. Eichstädt, A. Link, P. Harris and C. ElsterEfficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurementsMetrologia 49, 401-4102012
S. Nevas, G. Wübbeler, A. Sperling, C. Elster, and A. TeuberSimultaneous correction of bandpass and stray-light effects in array spectroradiometer dataMetrologia 49, 43-472012
C. Matthews, F. Pennecchi, S. Eichstädt, A. Malengo, T. Esward, I. Smith, C. Elster, A. Knott, F. Arrhén and A. LakkaMathematical modelling to support tracable dynamic calibration of pressure sensorsMetrologia 51, 326-3382014
S. Eichstädt, V. Wilkens, A. Dienstfrey, P. Hale, B. Hughes and C. JarvisOn challenges in the uncertainty evaluation for time-dependent measurementsMetrologia 53(4)2016
A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, D. F. WilliamsMinimum-Phase Calibration of Sampling OscilloscopesMicrowave Theory and Techniques, IEEE Transactions on2006
S. EichstädtAnalysis of Dynamic Measurements - Evaluation of dynamic measurement uncertaintyPhD Thesis, TU Berlin2012
Livina, VN, Lohmann, G, Mudelsee, M, Lenton, TMForecasting the underlying potential governing the time series of a dynamical systemPhysica A: Statistical Mechanics and its Applications2013
P. D. Hale, D. F. Williams, A. Dienstfrey, J. Wang, J. Jargon, D. Humphreys, M. Harper, H. Füser, M. BielerTraceability of High-Speed Electrical Waveforms at NIST, NPL, and PTBPrecision Electromagnetic Measurements (CPEM), 2012 Conference on2012
D. A. Humphreys, P. M. Harris, J. M. MiallInstrument related structure in covariance matrices used for uncertainty propagationProceedings of the 42nd European Microwave Conference2012
S. M. RiadThe deconvolution problem: An overviewProceedings of the IEEE1986
TJ Esward, C Elster, JP HesslingAnalysis of dynamic measurements: New challenges require new solutionsProceedings of XIX IMEKO World Congress, Lisbon, Portugal2009
S. Eichstädt, A. Link and C. ElsterDynamic uncertainty for compensated second-order systemsSensors 10, 7621-76312010
V. Wilkens, C. KochAmplitude and phase calibration of hydrophones up to 70 MHz using broadband pulse excitation and an optical reference hydrophoneThe Journal of the Acoustical Society of America2004
S. Eichstädt, T. J. Esward and A. SchäferOn the necessity of dynamic calibration for improved traceability of mechanical quantitiesXXI IMEKO World Congress, Prague, Czech Republic2015




Workshop Series

As part of the EURAMET TC-1078, the Physikalisch-Technische Bundesanstalt, the National Physical Laboratory (GB) and the Laboratoire national de métrologie et d'essais (France) organise the workshop series "Analysis of Dynamic Measurements"
  1. "Signal processing awareness seminar", NPL, UK, 2006
  2. "Analysis of dynamic measurements";, PTB, Germany, 2007
  3. "Analysis of dynamic measurements", NPL, UK, 2008
  4. "Session TC21- Dynamical Measurements" at IMEKO XIX World Congress, Portugal, 2009
  5. "5th workshop on the analysis of dynamic measurements", SP, Sweden, 2010
  6. "6th workshop on the analysis of dynamic measurements", Chalmers University, Sweden 2011
  7. "7th workshop on the analysis of dynamic measurements”, LNE, France, 2012
  8. "8th workshop on the analysis of dynamic measurements" INRIM, Italy, 2014.
  9. "9th International workshop on analysis of dynamic measurements", PTB Berlin, Germany 2016
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