This paper focuses on the mathematical modelling required to support the development of new primary standard systems for traceable calibration of dynamic pressure sensors. We address two fundamentally different approaches to realising primary standards, specifically the shock tube method and the drop-weight method. Focusing on the shock tube method, the paper presents first results of system identification and discusses future experimental work that is required to improve the mathematical and statistical models. We use simulations to identify differences between the shock tube and drop-weight methods, to investigate sources of uncertainty in the system identification process and to assist experimentalists in designing the required measuring systems. We demonstrate the identification method on experimental results and draw conclusions.
Analysis of dynamic measurements is of growing importance in metrology as an increasing number of applications requires the determination of measurands showing a time-dependence. Often linear time-invariant (LTI) systems are appropriate for modelling the relation between the available measurement data and the required time-dependent values of the measurand. Estimation of the measurand is then carried out by deconvolution.This paper is a tutorial about the application of digital deconvolution filters to reconstruct a time-variable measurand from the measurement signal of a LTI measurement apparatus. The goal of the paper is to make metrologists aware of the potentialities of digital signal processing in such cases. A range of techniques is available for the construction of a digital deconvolution filter. Here we compare various approaches for a form of dynamic model that is relevant to many metrological applications and we discuss the consequences for these approaches of the different ways in which information about the LTI system may be expressed. We consider specifically the methods of minimum-phase all pass decomposition, asynchronous time reversal using the exact inverse filter and the construction of stable infinite impulse response and finite impulse response approximate inverse filters by a least squares approach in the frequency domain. The methods are compared qualitatively by assessing their numerical complexity and quantitatively in terms of their performance for a simulated measurement task.Taking into account numerical complexity and underlying assumptions of the methods, we conclude that when a continuous model of the LTI system is available, or when the starting point is a set of measurements of the frequency response of a system, application of least squares in the frequency domain for the construction of an approximate inverse filter is to be preferred. On the other hand, asynchronous time reversal filtering using the exact inverse filter appears superior when a discrete model of the LTI system is available and when causality of the deconvolution filter is not an issue.
We have performed an optoelectronic measurement of the impulse response of an ultrafast sampling oscilloscope with a nominal bandwidth of 100 GHz within a time window of approximately 100 ps. Our experimental technique also considers frequency components above the cutoff frequency of higher-order modes of the 1.0 mm coaxial line, which is shown to be important for the specification of the impulse response of ultrafast sampling oscilloscopes. Additionally, we have measured the reflection coefficient of the sampling head induced by the mismatch of the sampling circuit and the coaxial connector which is larger than 0.5 for certain frequencies. The uncertainty analysis has been done using the Monte Carlo method of Supplement 1 to the 'Guide to the Expression of Uncertainty in Measurement' and correlations in the estimated impulse response have been determined. Our measurements extend previous work which deals with the characterization of 70 GHz oscilloscopes and the measurement of 100 GHz oscilloscopes up to the cutoff frequency of higher-order modes.
Measurement of quantities having time-dependent values such as force, acceleration or pressure is a topic of growing importance in metrology. The application of the Guide to the Expression of Uncertainty in Measurement (GUM) and its Supplements to the evaluation of uncertainty for such quantities is challenging. We address the efficient implementation of the Monte Carlo method described in GUM Supplements 1 and 2 for this task. The starting point is a time-domain observation equation. The steps of deriving a corresponding measurement model, the assignment of probability distributions to the input quantities in the model, and the propagation of the distributions through the model are all considered. A direct implementation of a Monte Carlo method can be intractable on many computers since the storage requirement of the method can be large compared with the available computer memory. Two memory-efficient alternatives to the direct implementation are proposed. One approach is based on applying updating formulae for calculating means, variances and point-wise histograms. The second approach is based on evaluating the measurement model sequentially in time. A simulated example is used to compare the performance of the direct and alternative procedures.
Bandpass correction in spectrometer measurements using monochromators is often necessary in order to obtain accurate measurement results. The classical approach of spectrometer bandpass correction is based on local polynomial approximations and the use of finite differences. Here we compare this approach with an extension of the Richardson-Lucy method, which is well known in image processing, but has not been applied to spectrum bandpass correction yet. Using an extensive simulation study and a practical example, we demonstrate the potential of the Richardson-Lucy method. In contrast to the classical approach, it is robust with respect to wavelength step size and measurement noise. In almost all cases the Richardson-Lucy method turns out to be superior to the classical approach both in terms of spectrum estimate and its associated uncertainties.
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