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Analysis of dynamic measurements

Working Group 8.42


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.

The typical scenario in a dynamic measurement is a time-dependent input signal, a linear time-invariant (LTI) measurement system and a corresponding time-dependent ouptut signal. The linearity of the system refers to the superposition of input signals, and the time-invariance ensures that the system itself does not change over time. Mathematically, the relation between input signal and output signal is in this case given by a convolution
$$ x(t) = \int y(\tau) h(\tau-t) d\tau , $$
with $y(t)$ the input signal (measurand), $h(t)$ the system's impulse response, and $x(t)$ the system's output signal. The measurand is then the input signal $x(t)$ of the measuring system, and its estimation requires deconvolution.

Statistical modeling of static quantities is realized by univariate or multivariate random variables. The evaluation of uncertainty is then based on probability density functions. In a mathematical sense, the extension of this concept to dynamic measurements requires the utilization of stochastic processes as a model for the uncertain knowledge of the continuous functions $x(t)$ and $y(t)$. That is, the dynamic quantity $x(t)$ is considered as a continuous function in time and uncertain knowledge about its values is expressed by a stochastic process $X_t$ with continuous trajectories. The working group has developed a corresponding framework for the evaluation of uncertainties which is consistent with the framework of the GUM for static measurements.

However, instead of the continuous function $x(t)$, typically its discretization is considered in the application. Therefore, the continuous output signal of the measurement system is stored on a computer as a vector, whereby $\mathbf{x}=(x_1,\ldots,x_N)$ with $x_k=x(t_k)$. Statistical modeling of the uncertain knowledge of the values of such a vector and its propagation can then, in principle, be carried out by using the concepts and framework of GUM Supplement 2. However, this is possible only for measurements on closed time intervals. Thus, it is not possible to treat running (on-time) measurements without making further assumptions.

A characteristic property of a dynamic measurement is that the output signal is not proportional to the input signal. This is due to so-called dynamic effects caused by the imperfect dynamic behavior of the measuring system. For instance, accelerometers typically show a resonance behavior, and hence, for a measured acceleration with a certain frequency content the output signal of the accelerometer shows a significant "ringing".

Fig. 1: Typical dynamic measurement with time-dependent errors in the output signal caused by the dynamic behavior of the measurement system.
Fig. 2: 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 other effects. In contrast to static measurements, this cannot be accomplished by scaling and shifting the output signal, since it only takes into account the static characteristics of the measuring system. Instead, for linear time-invariant systems (LTI) a so-called deconvolution has to be carried out. This allows the compensation of dynamic effects and, thereby, in principle, the reconstruction of the values of the dynamic measurand.

Fig. 3: Difference between the (time shifted) output signal and input signal from Fig. 1 with and without application of the compensation filter from Fig. 2



The analysis of dynamic measurements in the case of linear time-invariant (LTI) systems is typically 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 compensating its dynamic effects. 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.

Fig. 4: 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-dependent quantities. However, from a mathematical perspective the definition of a dynamic measurement can be extended to other independent quantities than time. This includes quantities whose value depends 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 a dynamic quantity if its value depends on another, independent quantity. A measurement is dynamic when at least one of the involved quantities is dynamic.

Fig. 5: Actual and measured spectral power distribution of a light source


The extended definition of a dynamic measurement contains a broad spectrum of metrological applications. Typical examples are measurements of mechanical quantities as, for example, force, torque and pressure. Further examples are oscilloscope measurements for the characterization of high speed electronics, hydrophone measurements for the characterization of medical ultrasound devices or the spectral characterization of radiation sources. The extended definition also includes, for example, the spectral characterization of luminous sources, spectral color measurements or camera-aided temperature measurements. The applications range from single sensor measurements up to large sensor networks.

In many areas of application, dynamic properties and dynamic errors have been neglected so far. Instead, rule-of-thumb correction methods were applied and larger uncertainties assigned to the measurement result. However, in recent years the demand for more precise dynamic measurements with smaller uncertainties has increased steadily.

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The PTB Working Group 8.42 has been carrying out research in the field of dynamic measurements for more than 10 years and its scientific work covers almost all areas of dynamic metrology. Publications of the working group contain, 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 filters. Moreover, the working group works closely together with industrial partners to promote developments in dynamic metrology.

The mathematical and statistical treatment of dynamic measurements requires different approaches and tools than the analysis of static measurements. Thus, the areas of research in the working group are determined by the challenges in dynamic metrology.

Propagation of uncertainties

In general, the 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 often determined by repeated measurements. Therefore, for univariate quantities a rather small number of measurements are 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 data sets, with a typical dynamic measurement consisting of more than one 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 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 to suppress undesired high frequency components. 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, this is only possible by taking into account prior knowledge about the measurand, which 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 any case, the low-pass filter causes a systematic deviation in the estimation result. A reduction in the systematic deviation is always attended by an increase in the noise-dependent variance in the result of deconvolution and vice verse. For metrological applications, these systematic errors have to be considered in the uncertainty budget. However, so far no general guidelines or harmonized treatment of these uncertainty contributions is available.

Practical challenges

The treatment of dynamic measurements leads to a number of practical challenges for metrologists. One of the currently most urging challenges is that of transferring the measurement result. Owing to the high dimensionality of discretized dynamic measurements, the associated uncertainties are also high-dimensional. For instance, the transfer of the calibration results for the impulse response of a sampling oscilloscope requires transferring a covariance matrix of the dimension of at least $1000\times 1000$. This is not possible with the current practice of paper-based calibration certificates.

Dynamic calibration

A large number of guidelines and standards are available for carrying out and taking advantage of static calibration. The analysis of dynamic measurements, though, is based on a dynamic calibration of the measuring system. The realization of dynamic calibration requires completely different measurement approaches and mathematical tools than static calibration. In some areas first attempts have been made, but in general this constitutes a significant challenge for future research.

For the treatment of the above-mentioned challenges, the working group has developed mathematical methodologies for the

  • GUM-compliant evaluation of dynamic calibration experiments,
  • design of digital deconvolution filters for the compensation of dynamic errors,
  • GUM-compliant evaluation of uncertainties for the application of digital filters,
  • efficient implementation of GUM Monte Carlo for the application of digital filters,
  • iterative estimation of spectral power distributions,
  • regularized deconvolution, including GUM-compliant uncertainty evaluation.
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In order to facilitate the application of the methods developed in the working group, suitable methods are available as free software implementations. For questions, remarks and suggestions, please contact Opens window for sending emailSascha Eichstädt.

PyDynamic - Software for the analysis of dynamic measurements

As part of the EMPIR project 14SIP08 Dynamic the PTB working group 8.42 together with the National Physical Laboratory (UK) develops a comprehensive Python software package, which contains a large variety of methods for the analysis of dynamic measurements.

Opens external link in new windowSoftware download

Monte Carlo for dynamic measurements

The propagation of measurement uncertainties for dynamic measurements using the Monte Carlo propagation of distributions requires efficient implementation in order to achieve a high degree of accuracy, since the computer memory requirement for a standard implementation is very high. In the working group 8.42 a MATLAB software package has been developed to carry out Monte Carlo for dynamic measurements with high accuracy on standard desktop computers.

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related publication

S. Eichstädt, A. Link, P. Harris and C. Elster (2012). Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements. Metrologia 49, 401-410  
(doi:Opens external link in new window10.1088/0026-1394/49/3/401).

Spectral deconvolution with Richardson-Lucy

The correction of deviations in spectra measured with a spectrometer is often necessary in order to obtain accurate results. The classical approach for such a correction is based on a method by Stearns. However, it has been demonstrated that the Richardson-Lucy method can produce much better results here. Therefore, the PTB Working Group has written a software implementation of an adapted Richardson-Lucy method with an automatic stopping rule. This software includes MATLAB code, a Python code as well as a graphical user interface written in Python.

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related publication

S. Eichstädt F. Schmähling G. Wübbeler, K. Anhalt, L. Bünger, K. Krüger and C. Elster (2013). Comparison of the Richardson-Lucy method and a classical approach for spectrometer bandpass correction. Metrologia 50, 107-118 (doi: 10.1088/0026-1394/50/2/107)

Evaluation of uncertainties in the application of the DFT

The Fourier transform and its counterpart for discrete time signals, the DFT, are common tools in measurement science and application. Although almost every scientific software package offers ready-to-use implementations of the DFT, the propagation of uncertainties in line with GUM is typically neglected. This is of particular importance in dynamic metrology, when input estimation is carried out by deconvolution in the frequency domain. To this end, we present the new open-source software tool GUM2DFT, which utilizes closed formulas for the efficient propagation of uncertainties for the application of the DFT, inverse DFT and input estimation in the frequency domain. It handles different frequency domain representations, accounts for autocorrelation and takes advantage of the symmetry inherent in the DFT result for real-valued time domain signals.

Opens internal link in current windowSoftware download

related publication

S. Eichstädt and V. Wilkens "GUM2DFT -- A software tool for uncertainty evaluation of transient signals in the frequency domain". Meas. Sci. Technol. 27(5), 055001, 2016 (doi: 10.1088/0957-0233/27/5/055001)

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Together with the National Physical Laboratory (UK) and the Laboratoire national de métrologie et d'essais (France), the working group is organizing the workshop series "Analysis of Dynamic Measurements" as part of Opens external link in new windowEURAMET TC-1078.

  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. Opens external link in new windowSession 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. "Opens external link in new window7th 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 the analysis of dynamic measurements", PTB, Germany, 2016
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running projects

completed projects

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Publication single view

Contribution to book

Title: Analysis of dynamic measurements: compensation of dynamic error and evaluation of uncertainty
Author(s): C. Elster and A. Link
Editor: F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang
Year: 2009
Volume: Advanced Mathematical & Computational Tools in Metrology VIII
Series: Series on Advances in Mathematics for Applied Sciences
Edition: 78
Chapter: 13
Pages: 80-89
World Scientific New Jersey
Tags: 8.42, Dynamik, Unsicherheit

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