Deconvolution filters for the analysis of dynamic measurement processes: a tutorial

Abstract

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.

An application of system identification in metrology

“…Trevor Esward National Physics Laboratory, London) pointed the link of the reported research to the literature on the dynamic measurement problem (Eichstädt et al 2010). The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007(FP7/ -2013 / ERC Grant agreement number 258581 "Structured low-rank approximation: Theory, algorithms, and applications".…”

“…The classical approach in dynamic measurement assumes that the process dynamics is known and linear timeinvariant, see (Eichstädt et al 2010) for an overview. Consequently, the compensator is also a linear time-invariant system, designed by frequency or time domain de-convolution techniques.…”

Evaluation of measurement uncertainty for time-dependent quantities

“…We assume that the device can be modelled as a linear time-invariant system [6]. The goal is to design a digital compensation filter that approximates the inverse of this LTI system up to a certain frequency and attenuates measurement noise beyond that frequency [11]. Therefore, we consider a cascade of an inverse filter ݃ ௩ ሺ‫ݐ‬ሻ and a low-pass filter ݃ ௪ ሺ‫ݐ‬ሻ with frequency response ‫ܩ‬ ௩ ሺ݁ ఠ் ೞ ሻ and ‫ܩ‬ ௪ ሺ݁ ఠ் ೞ ሻ, respectively [7].…”

“…The inverse filter ݃ ௩ ሺ‫ݐ‬ሻ is designed from knowledge about the measurement device. This may be available in terms of a parametric model, a sequence of frequency response values or a sequence of discrete-time impulse response values [6,11]. Let us consider, for instance, that a set of frequency response values ൌ ൫‫ܪ‬ሺ݆߱ ଵ ሻǡ ǥ ǡ ‫ܪ‬ሺ݆߱ ெ ሻ൯ ் is available in terms of real and imaginary parts ൌ ሺܴ݁ሾ ் ሿǡ ‫݉ܫ‬ሾ ሿሻ ் and their associated uncertainties.…”

“…This is the case when, for instance, a sensor is calibrated using sinusoidal excitations over a certain range of frequencies. Consider furthermore, that a time delay of ݊ ௗ samples is introduced for in order to improve the estimation quality [11][12][13]. Hence, the time-shifted response of the measurement device with frequency response values ‫ܪ‬ ෩ ሺ߱ሻ ൌ ‫ܪ‬ሺ݆߱ሻ݁ ఠ ் ೞ is considered.…”