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Analysis of dynamic measurements
Description
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].
Typical dynamic measurement with time dependent errors in the output signal caused by the dynamic behavior of the measurement system.
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].
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].
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].
Difference between the (time shifted) output signal and the above input signal with and without application of the compensation filter
Actual and measured spectral power distribution of a light source
References
- A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, D. F. Williams. Minimum-Phase Calibration of Sampling Oscilloscopes. Microwave Theory and Techniques, IEEE Transactions on, 2006
- S. Eichstädt, C. Elster, T. J. Esward and J. P. Hessling. Deconvolution filters for the analysis of dynamic measurements: a tutorial. Metrologia 47, 522-533, 2010
- S. Eichstädt, F. Schmähling, G. Wübbeler, K. Anhalt, L. Bünger, U. Krüger and C. Elster. Comparison of the Richardson-Lucy method and a classical approach for spectrometer bandpass correction. Metrologia 50, 107-118, 2013
- H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschke, K. Pierz and M. Bieler. Optoelectronic time-domain characterization of a 100 GHz sampling oscilloscope. Meas. Sci. Technol. 23, 025201, 2012
- P. D. Hale, D. F. Williams, A. Dienstfrey, J. Wang, J. Jargon, D. Humphreys, M. Harper, H. Füser, M. Bieler. Traceability of High-Speed Electrical Waveforms at NIST, NPL, and PTB. Precision Electromagnetic Measurements (CPEM), 2012 Conference on, 2012
- L. Klaus, B. Arendacká, M. Kobusch and T. Bruns. Dynamic torque calibration by means of model parameter identification. ACTA IMEKO, 2015
- M. Kobusch, S. Eichstädt, L. Klaus, T. Bruns. Investigations for the model-based dynamic calibration of force transducers by using shock excitation. ACTA IMEKO, 2015
- A. Link, C. Elster. Uncertainty evaluation for IIR (infinite impulse response) filtering using a state-space approach. Metrologia, 2009
- S. M. Riad. The deconvolution problem: An overview. Proceedings of the IEEE, 1986
- C. Schlegel, G. Kieckenap, B. Glöckner, A. Buß and R. Kumme . Traceable periodic force calibration. Metrologia, 2012
- K. A. Wear, Y. Liu, P. M. Gammell, S. Maruvada, and G. R. Harris. Correction for Frequency-Dependent Hydrophone Response to Nonlinear Pressure Waves Using Complex Deconvolution and Rarefactional Filtering: Application With Fiber Optic Hydrophones Keith. IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, 2015
- V. Wilkens, C. Koch. Amplitude and phase calibration of hydrophones up to 70 MHz using broadband pulse excitation and an optical reference hydrophone. The Journal of the Acoustical Society of America, 2004
Research
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.
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
Authors
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Title
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Journal
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Year
|
|---|---|---|---|
| A. Dienstfrey, P. D. Hale | Colored Noise and Regularization Parameter Selection for Waveform Metrology | Instrumentation and Measurement, IEEE Transactions on | 2014 |
| A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, D. F. Williams | Minimum-Phase Calibration of Sampling Oscilloscopes | Microwave Theory and Techniques, IEEE Transactions on | 2006 |
| A. Link, C. Elster | Uncertainty evaluation for IIR (infinite impulse response) filtering using a state-space approach | Metrologia | 2009 |
| B. Arendacká, A. Täubner, S. Eichstädt, T. Bruns and C. Elster | Linear mixed models: GUM and beyond | Meas. Sci. Rev. 14, 52-61 | 2012 |
| C. Elster, A. Link | Uncertainty evaluation for dynamic measurements modelled by a linear time-invariant system | Metrologia | 2008 |
| C. Matthews, F. Pennecchi, S. Eichstädt, A. Malengo, T. Esward, I. Smith, C. Elster, A. Knott, F. Arrhén and A. Lakka | Mathematical modelling to support tracable dynamic calibration of pressure sensors | Metrologia 51, 326-338 | 2014 |
| C. Schlegel, G. Kieckenap, B. Glöckner, A. Buß and R. Kumme | Traceable periodic force calibration | Metrologia | 2012 |
| Collett M, Esward T J, Harris P M, Matthews C E, Smith I M | Simulating distributed measurement networks in which sensors may be faulty, noisy and interdependent: A software tool for sensor network design, data fusion and uncertainty evaluation | Measurement | 2013 |
| D. A. Humphreys, P. M. Harris, J. M. Miall | Instrument related structure in covariance matrices used for uncertainty propagation | Proceedings of the 42nd European Microwave Conference | 2012 |
| D.A. Humphreys, P.M. Harris, M. Rodriguez-Higuero, F.A. Mubarak, D. Zhao and K. Ojasalo | Principal component compression method for covariance matrices used for uncertainty propagation | IEEE Transactions on Instrumentation and Measurement, vol 64(2) | 2014 |
| G. H. Nam, M. G. Cox, P. M. Harris, S. P. Robinson, G. Hayman, G. A. Beamiss, T. J. Esward and I. M. Smith | A model for characterizing the frequency-dependent variation in sensitivity with temperature of underwater acoustic transducers from historical calibration data | Meas. Sci. Technol. 18, 1553-1562 | 2007 |
| H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschke, K. Pierz and M. Bieler | Optoelectronic time-domain characterization of a 100 GHz sampling oscilloscope | Meas. Sci. Technol. 23, 025201 | 2012 |
| J. P. Hessling | A novel method of estimating dynamic measurement errors | Measurement Science and Technology | 2006 |
| J. P. Hessling | A novel method of dynamic correction in the time domain | Measurement Science and Technology | 2008 |
| J. P. Hessling | Dynamic metrology—an approach to dynamic evaluation of linear time-invariant measurement systems | Measurement Science and Technology | 2008 |
| J. P. Hessling | A novel method of evaluating dynamic measurement uncertainty utilizing digital filters | Measurement Science and Technology, 20, nr. 5, 055106 | 2009 |
| J. P. Hessling | Dynamic calibration of uni-axial material testing machines | Mechanical systems and signal processing, 22, nr. 2, 451-466 | 2008 |
| K. A. Wear, Y. Liu, P. M. Gammell, S. Maruvada, and G. R. Harris | Correction for Frequency-Dependent Hydrophone Response to Nonlinear Pressure Waves Using Complex Deconvolution and Rarefactional Filtering: Application With Fiber Optic Hydrophones Keith | IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL | 2015 |
| L. Klaus, B. Arendacká, M. Kobusch and T. Bruns | Dynamic torque calibration by means of model parameter identification | ACTA IMEKO | 2015 |
| Livina, VN, Lohmann, G, Mudelsee, M, Lenton, TM | Forecasting the underlying potential governing the time series of a dynamical system | Physica A: Statistical Mechanics and its Applications | 2013 |
| M. Kobusch, S. Eichstädt, L. Klaus, T. Bruns | Investigations for the model-based dynamic calibration of force transducers by using shock excitation | ACTA IMEKO | 2015 |
| M. Musi? and M. Ahi?-Džoki? and Z. Džemi? | A new approach to detection of vortices using ultrasound | Flow Measurement and Instrumentation | 2015 |
| P. D. Hale, A. Dienstfrey, J. Wang, D. F. Williams, A. Lewandowski, D. A. Keenan, T. S. Clement | Traceable Waveform CalibrationWith a Covariance-Based Uncertainty Analysis | Instrumentation and Measurement, IEEE Transactions on | 2009 |
| P. D. Hale, D. F. Williams, A. Dienstfrey, J. Wang, J. Jargon, D. Humphreys, M. Harper, H. Füser, M. Bieler | Traceability of High-Speed Electrical Waveforms at NIST, NPL, and PTB | Precision Electromagnetic Measurements (CPEM), 2012 Conference on | 2012 |
| S. Eichstädt | Analysis of Dynamic Measurements - Evaluation of dynamic measurement uncertainty | PhD Thesis, TU Berlin | 2012 |
| S. Eichstädt and C. Elster | Reliable uncertainty evaluation for ODE parameter estimation - a comparison | J. Phys. 490, 1, 012230 | 2014 |
| S. Eichstädt and C. Elster | Uncertainty evaluation for continuous-time measurements | Advanced Mathematical and Computational Tools in Metrology and Testing IX | 2012 |
| S. Eichstädt and V. Wilkens | GUM2DFT - A software tool for uncertainty evaluation of transient signals in the frequency domain | Meas. Sci. and Technol., 27(5) | 2016 |
| S. Eichstädt, A. Link and C. Elster | Dynamic uncertainty for compensated second-order systems | Sensors 10, 7621-7631 | 2010 |
| S. Eichstädt, A. Link, P. Harris and C. Elster | Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements | Metrologia 49, 401-410 | 2012 |
| S. Eichstädt, A. Link, T. Bruns and C. Elster | On-line dynamic error compensation of accelerometers by uncertainty-optimal filtering | Measurement 43, 708-713 | 2010 |
| S. Eichstädt, B. Arendacká, A. Link and C. Elster | Evaluation of measurement uncertainties for time-dependent quantities | EPJ Web of Conferences 77 | 2014 |
| S. Eichstädt, C. Elster, I.M. Smith, T.J. Esward | Evaluation of dynamic measurement uncertainty – an open-source software package to bridge theory and practice. | J. Sens. Sens. Syst., 6 97-105 | 2017 |
| S. Eichstädt, C. Elster, T. J. Esward and J. P. Hessling | Deconvolution filters for the analysis of dynamic measurements: a tutorial | Metrologia 47, 522-533 | 2010 |
| S. Eichstädt, N. Makarava and C. Elster | On the evaluation of uncertainties for state estimation with the Kalman filter | Meas. Sci. Technol. vol. 27(12), 125009, 2016 | 2016 |
| S. Eichstädt, T. J. Esward and A. Schäfer | On the necessity of dynamic calibration for improved traceability of mechanical quantities | XXI IMEKO World Congress, Prague, Czech Republic | 2015 |
| S. Eichstädt, V. Wilkens, A. Dienstfrey, P. Hale, B. Hughes and C. Jarvis | On challenges in the uncertainty evaluation for time-dependent measurements | Metrologia 53(4) | 2016 |
| S. M. Riad | The deconvolution problem: An overview | Proceedings of the IEEE | 1986 |
| S. Nevas, G. Wübbeler, A. Sperling, C. Elster, and A. Teuber | Simultaneous correction of bandpass and stray-light effects in array spectroradiometer data | Metrologia 49, 43-47 | 2012 |
| T. Bruns, A. Link and A. Täubner | The influence of different vibration exciter systems on high frequency primary calibration of single-ended accelerometers | Metrologia 49, 27-31 | 2012 |
| T. Esward, C. Matthews, S. Downes, A. Knott, S. Eichstädt and C. Elster | Uncertainty evaluation for traceable dynamic measurement of mechanical quantities: A case study in dynamic pressure calibration | Advanced 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 Jersey | 2012 |
| TJ Esward, C Elster, JP Hessling | Analysis of dynamic measurements: New challenges require new solutions | Proceedings of XIX IMEKO World Congress, Lisbon, Portugal | 2009 |
| V. Wilkens, C. Koch | Amplitude and phase calibration of hydrophones up to 70 MHz using broadband pulse excitation and an optical reference hydrophone | The Journal of the Acoustical Society of America | 2004 |
Links
Projects
- EMPIR 14SIP08 "Standards and software to maximise end user uptake of NMI calibrations of dynamic force, torque and pressure sensors" (05/2015 - 04/2018)
- EMRP ENG63 "Sensor network metrology for the determination of electrical grid characteristics" (07/2014 - 06/2017)
- EMRP IND09 project "Traceable dynamic measurement of mechanical quantities" (09/2011 - 08/2014)
- EURAMET TC-1078 Development of methods for the evaluation of uncertainty in dynamic measurements
Other
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"
- "Signal processing awareness seminar", NPL, UK, 2006
- "Analysis of dynamic measurements";, PTB, Germany, 2007
- "Analysis of dynamic measurements", NPL, UK, 2008
- "Session TC21- Dynamical Measurements" at IMEKO XIX World Congress, Portugal, 2009
- "5th workshop on the analysis of dynamic measurements", SP, Sweden, 2010
- "6th workshop on the analysis of dynamic measurements", Chalmers University, Sweden 2011
- "7th workshop on the analysis of dynamic measurements”, LNE, France, 2012
- "8th workshop on the analysis of dynamic measurements" INRIM, Italy, 2014.
- "9th International workshop on analysis of dynamic measurements", PTB Berlin, Germany 2016