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 timeinvariant 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 timeinvariant (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 $$x(t) = (y\ast h)(t) = \int_{\infty}^{\infty} y(s)h(st)ds $$.
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, highspeed electronics, medical ultrasound, 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 nonnegative 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. MinimumPhase 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, 522533, 2010
 S. Eichstädt, F. Schmähling, G. Wübbeler, K. Anhalt, L. Bünger, U. Krüger and C. Elster. Comparison of the RichardsonLucy method and a classical approach for spectrometer bandpass correction. Metrologia 50, 107118, 2013
 H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschke, K. Pierz and M. Bieler. Optoelectronic timedomain 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 HighSpeed 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 modelbased 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 statespace 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 FrequencyDependent 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


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 frequencydependent variation in sensitivity with temperature of underwater acoustic transducers from historical calibration data  Meas. Sci. Technol. 18, 15531562  2007 
M. Music and M. AhicDžokic and Z. Džemic  A new approach to detection of vortices using ultrasound  Flow Measurement and Instrumentation  2015 
J. P. Hessling  A novel method of dynamic correction in the time domain  Measurement Science and Technology  2008 
J. P. Hessling  A novel method of estimating dynamic measurement errors  Measurement Science and Technology  2006 
J. P. Hessling  A novel method of evaluating dynamic measurement uncertainty utilizing digital filters  Measurement Science and Technology, 20, nr. 5, 055106  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 
S. Eichstädt  Analysis of Dynamic Measurements  Evaluation of dynamic measurement uncertainty  PhD Thesis, TU Berlin  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 
A. Dienstfrey, P. D. Hale  Colored Noise and Regularization Parameter Selection for Waveform Metrology  Instrumentation and Measurement, IEEE Transactions on  2014 
K. A. Wear, Y. Liu, P. M. Gammell, S. Maruvada, and G. R. Harris  Correction for FrequencyDependent 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 
S. Eichstädt, C. Elster, T. J. Esward and J. P. Hessling  Deconvolution filters for the analysis of dynamic measurements: a tutorial  Metrologia 47, 522533  2010 
J. P. Hessling  Dynamic calibration of uniaxial material testing machines  Mechanical systems and signal processing, 22, nr. 2, 451466  2008 
J. P. Hessling  Dynamic metrology—an approach to dynamic evaluation of linear timeinvariant measurement systems  Measurement Science and Technology  2008 
L. Klaus, B. Arendacká, M. Kobusch and T. Bruns  Dynamic torque calibration by means of model parameter identification  ACTA IMEKO  2015 
S. Eichstädt, A. Link and C. Elster  Dynamic uncertainty for compensated secondorder systems  Sensors 10, 76217631  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, 401410  2012 
S. Eichstädt, C. Elster, I.M. Smith, T.J. Esward  Evaluation of dynamic measurement uncertainty – an opensource software package to bridge theory and practice.  J. Sens. Sens. Syst., 6 97105  2017 
S. Eichstädt, B. Arendacká, A. Link and C. Elster  Evaluation of measurement uncertainties for timedependent quantities  EPJ Web of Conferences 77  2014 
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 
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 
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 
M. Kobusch, S. Eichstädt, L. Klaus, T. Bruns  Investigations for the modelbased dynamic calibration of force transducers by using shock excitation  ACTA IMEKO  2015 
B. Arendacká, A. Täubner, S. Eichstädt, T. Bruns and C. Elster  Linear mixed models: GUM and beyond  Meas. Sci. Rev. 14, 5261  2012 
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, 326338  2014 
A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, D. F. Williams  MinimumPhase Calibration of Sampling Oscilloscopes  Microwave Theory and Techniques, IEEE Transactions on  2006 
S. Eichstädt, V. Wilkens, A. Dienstfrey, P. Hale, B. Hughes and C. Jarvis  On challenges in the uncertainty evaluation for timedependent measurements  Metrologia 53(4)  2016 
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, A. Link, T. Bruns and C. Elster  Online dynamic error compensation of accelerometers by uncertaintyoptimal filtering  Measurement 43, 708713  2010 
H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschke, K. Pierz and M. Bieler  Optoelectronic timedomain characterization of a 100 GHz sampling oscilloscope  Meas. Sci. Technol. 23, 025201  2012 
D.A. Humphreys, P.M. Harris, M. RodriguezHiguero, 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 
S. Eichstädt and C. Elster  Reliable uncertainty evaluation for ODE parameter estimation  a comparison  J. Phys. 490, 1, 012230  2014 
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 
S. Nevas, G. Wübbeler, A. Sperling, C. Elster, and A. Teuber  Simultaneous correction of bandpass and straylight effects in array spectroradiometer data  Metrologia 49, 4347  2012 
S. M. Riad  The deconvolution problem: An overview  Proceedings of the IEEE  1986 
T. Bruns, A. Link and A. Täubner  The influence of different vibration exciter systems on high frequency primary calibration of singleended accelerometers  Metrologia 49, 2731  2012 
P. D. Hale, D. F. Williams, A. Dienstfrey, J. Wang, J. Jargon, D. Humphreys, M. Harper, H. Füser, M. Bieler  Traceability of HighSpeed Electrical Waveforms at NIST, NPL, and PTB  Precision Electromagnetic Measurements (CPEM), 2012 Conference on  2012 
C. Schlegel, G. Kieckenap, B. Glöckner, A. Buß and R. Kumme  Traceable periodic force calibration  Metrologia  2012 
P. D. Hale, A. Dienstfrey, J. Wang, D. F. Williams, A. Lewandowski, D. A. Keenan, T. S. Clement  Traceable Waveform CalibrationWith a CovarianceBased Uncertainty Analysis  Instrumentation and Measurement, IEEE Transactions on  2009 
S. Eichstädt and C. Elster  Uncertainty evaluation for continuoustime measurements  Advanced Mathematical and Computational Tools in Metrology and Testing IX  2012 
C. Elster, A. Link  Uncertainty evaluation for dynamic measurements modelled by a linear timeinvariant system  Metrologia  2008 
A. Link, C. Elster  Uncertainty evaluation for IIR (infinite impulse response) filtering using a statespace approach  Metrologia  2009 
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 
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 TC1078 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