The goal of dynamic measurements is the determination of a physical quantity whose value shows a timedependence. Examples are for instance the measurement of a mechanical force or acceleration, electrical highvoltage impulses or a timedependent temperature.
Ideally, a dynamic measurement system is designed such that its output signal is proportional to the input signal, i.e. the timedependent value of the measurand. However, often such an ideal behavior is achieved only for input signals with a lowfrequency spectrum. For input signals with large bandwidth, the output signal of the measurement system is no longer proportional to its input signal. By taking the output signal of the measurement system as a timedependent estimate of the value of the measurand, the socalled dynamic error is introduced. In order to compensate for the dynamic error, postprocessing of the output of the measurement system is required for which digital filtering is an appropriate tool. In order to design a compensation filter, the dynamic behavior of the measurement system needs to be characterized, i.e. the measurement system has to be identified.
The determination of measurement uncertainty plays a key role in metrology. For a measurand with a constant value, agreed defacto standards for the determination of measurement uncertainty are available. These standards are based on a Bayesian point of view and allow for a consistent treatment of random and systematic influences. However, these standards cannot be immediately applied to the case where the value of the measurand is timedependent, and the concepts applied for the static case need to be extended.
Currently, we focus our work on the development and application of methods for
Figure 1 shows schematically a dynamic measurement system with subsequent postprocessing. The timedependent value of the measurand passes through the measurement system and an analogue todigital conversion. Subsequent digital filtering compensates for the dynamic error and yields an estimate of the measurand including the dynamic uncertainty associated with this estimate.
Fig. 1: Schematic setup of dynamic measurement with subsequent compensation.
Actually, a compensation filter should have a frequency response which equals the inverse frequency response of the measurement system. However, this would imply a strong amplification of high frequency (noise) components. Therefore, the design of a compensation filter requires some tradeoff between sufficient noise suppression in the high frequency region and tolerable signal distortion due to nonideal inverse filtering. Figure 2 shows the magnitude frequency response of such a compensation filter for a particular measurement system. The frequency response of the compensated measurement system is constant up to 60 kHz.
Fig. 2: Magnitude plots of the frequency responses of a measurement system (black), the compensation filter (blue) and the compensated measurement system (green).
The need and the benefit of applying a compensation filter to this measurement system is illustrated for a particular test signal in figure 3 where also the dynamic error is shown if no compensation is applied.
Fig. 3: Input and output signals of a measurement system (top) as well as difference signal between the (timeshifted) output and input with (green) and without (red) applying the digital compensation filter of Fig. 2.
Figure 4 finally illustrates for simulated data the comparison of an uncertainty evaluation based on a static and a dynamic analysis for the above example. The static analysis treats the output of the measurement system as proportional to its input and ignores the dynamic behavior. The dynamic analysis is based on a subsequent application of the compensation filter. Figure 4 shows the frequency with which 95% credibility intervals determined by the analyses cover the underlying true value of the measurand at the time of the input signal’s maximum. The results show that as the bandwidth of the measurand increases, the coverage probability of 95% credibility intervals tends to zero for the static analysis in contrast to the dynamic analysis, and reliable uncertainties can in this case only be obtained by accounting for the dynamic behavior of the sensor system.
Fig. 4: Estimated coverage probability for 95 % credibility intervals obtained for the peaks of a Gaussianlike measurand using the static and dynamic approaches in the uncertainty analysis.
Monte Carlo  The propagation of measurement uncertainty in dynamic measurements requires an efficient implementation in order to achieve high accuracies. The reason is that a standard implementation requires a large amount of computer memory.
In the PTB working group 8.42 a MATLAB software package has been developed, which implements these methods in an easytouse way. This package can be downloaded for free. This software will be updated continuously. If you want to get informed about updates or have any questions or suggestions, please contact Sascha Eichstädt .  
RichardsonLucy  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, we demonstrated in
that the RichardsonLucy method can result in much better results. Therefore, a software implementation of an adapted RichardsonLucy method with automatic stopping rule has been written in PTB working group 8.42. This software includes MATLAB code as well as a graphical user interface written in Python. The software can be downloaded for free. This software will be updated continuously. If you want to get informed about updates or have any questions or suggestions, please contact Sascha Eichstädt .

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S. Eichstädt (2012). Analysis of Dynamic Measurements  Evaluation of dynamic measurement uncertainty. PhD Thesis. [download pdf (1MB)]
S. Eichstädt and C. Elster (2012). Uncertainty evaluation for continuoustime measurements, in "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.
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