# Analysis of dynamic measurements

Working Group 8.42

#### Content

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 require different approaches than the analysis of static measurements this part of metrology is often call Dynamic Metrology. The mathematical modelling 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 is with respect 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 output signal. Estimation of the measurand thus requires a deconvolution.

Statistical modelling of static quantities is realized by univariate or multivariate random variables. Evaluation of uncertainty is then based on probability density functions. Strictly speaking, 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.

However, typically instead of the continuous function $x(t)$ its discretization is considered in the application. Therefore, the continuous output signal of the measurement system is stored on a computer as a vector $\mathbf{x}=(x_1,\ldots,x_N)$ with $x_k=x(t_k)$. Statistical modelling and propagation of uncertainties can then, in principle, be carried out by using the concepts and framework of GUM Supplement 2. However, this is possible only on closed time intervals.

A characteristic property of a dynamic measurement is that the output signal is not proportional to the input signal. The reason for that are so called dynamic effects caused by the measurement 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".

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. 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 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.

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 than 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 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.

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.

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

# Research

The PTB working group 8.42 carries 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 is working 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, 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.

#### Practical challenges

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

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.

# Software

In order to make the application of the methods developed in the working group as easy as possible, suitable methods are available as free software implementations.For questions, remarks and suggestions contact Sascha Eichstädt.

Monte Carlo for dynamic measurements

The propagation of measurement uncertainties for dynamic measurements using the Monte Carlo propagation of distributions requires an efficient implementation in order to achieve sufficient accuracies. In the working group 8.42 a MATLAB software has been developed to carry out Monte Carlo for dynamic measurements on standard desktop computers with high accuracy.

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:10.1088/0026-1394/49/3/401).

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 from Stearns. However, it has been demonstrated that the Richardson-Lucy method can result in much better results. Therefore, a software implementation of an adapted Richardson-Lucy method with automatic stopping rule has been written in PTB working group 8.42. This software includes MATLAB code, Python code as well as a graphical user interface written in Python.

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)

# Workshops

Together with the National Physical Laboratory (UK) and the Laboratoire national de métrologie et d'essais (France), the working group organises the workshop series "Analysis of Dynamic Measurements" as part of EURAMET 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. 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

# Projects

running projects

finished projects

References