Analysis of dynamic measurements
Description
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].


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].
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
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
Propagation of uncertainties
Estimating the measurand
Related journal papers
Authors
|
Title
|
Journal
|
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. Music and M. Ahic-Džokic and Z. Džemic | 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
- "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