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The compensation of LTI systems and the evaluation of the according uncertainty is of growing interest in metrology. Uncertainty evaluation in metrology ought to follow specific guidelines, and recently two corresponding uncertainty evaluation schemes have been proposed for FIR and IIR filtering. We employ these schemes to compare an FIR and an IIR approach for compensating a second-order LTI system which has relevance in metrology. Our results suggest that the FIR approach is superior in the sense that it yields significantly smaller uncertainties when real-time evaluation of uncertainties is desired.

Various important types of sensors like accelerometers or load cells can be modeled by a mass-spring system resulting in a second-order model of the kind:
_{0}, _{0} = 2_{0} denote static gain, damping and resonance frequency, see [

The model parameters in (1) are usually not known from the start, but need to be determined by system identification using calibration measurements, see [

Metrology is another field with a recently growing interest in the compensation of uncertain dynamic systems [

For the particular model (1) recently two approaches have been proposed for the compensation of dynamic effects in terms of an IIR [

The goal of this paper is to compare the performance of the two particular approaches [

We consider the following measurement task: a continuous-time input signal _{S}_{S}_{S}

We consider the two recently proposed approaches [

We describe uncertainty evaluation in line with the GUM and briefly recall the two considered uncertainty evaluation methods for FIR and IIR filtering.

We assume that the characterization of the sensor in terms of calibration measurements provides parameter estimates _{0}, _{0} for the system (1) with an uncertainty matrix _{0}, _{0}), see [_{θ̂}_{θ̂}

In addition to _{θ̂}_{comp}[_{0} denotes a possible known time sample delay. Utilizing the well-known inequality for the Fourier transform _{S}_{S}^{jΩ/fS}

In order to determine the contribution of the dynamic errors to the uncertainty

The overall dynamic uncertainty is then evaluated according to:

For the evaluation of the uncertainty

For an uncertainty evaluation in the context of FIR filtering the variance term in (6) can be calculated in a straightforward way, see [_{low}[_{low}[_{low}[_{comp}])^{T}; _{low} denotes the low-pass filtered sensor output signal and _{ylow} stands for the covariance matrix of _{low}[

We compare the two compensation filter methods [

As input signal we chose a low-pass filtered rectangular function, where we employed low-pass filter cut-off frequencies of 10 kHz and 25 kHz to limit the bandwidth of the sensor input signal. The sensor output signal was calculated by a convolution of the chosen input signal with the LTI system transfer function (1) using the parameters in (11). ^{2} = 1 e−3, σ^{2} = 3 e−4, and σ^{2} = 1 e−6, respectively. As sampling frequency we chose 500 kHz. According to

The IIR deconvolution filter was derived according to [_{T}

We discretized this system employing the bilinear transform with frequency pre-warping to meet the resonance frequency, see [

The FIR deconvolution filter was designed according to [

A comparison of the frequency response of the compensated systems shows that both, FIR as well as IIR filter, yield a good approximation to the inverse of model (1) in the relevant frequency region for the available knowledge about the actual model parameters. While the phase response of the compensated system for the IIR filter is only approximately linear, the FIR filter results in a compensated system with an almost perfect linear phase response that can be realized in the time domain by a sample shift. Thus, the corresponding error bound (4) for the IIR compensation filter is larger than that for the FIR filter. This can be seen in

It appears that the shape of the uncertainties for the FIR and IIR compensation are similar. As expected, for both filter types a larger noise variance results in an increased uncertainty of the input signal estimate. The influence of the model uncertainty, namely the impact of the resonance frequency uncertainty _{0}) and damping uncertainty

It should be noted that the frequency responses of the compensated system shown in

An FIR and an IIR filter approach for the compensation of a second-order system have been compared in terms of resulting uncertainties. The main drawback of the considered IIR filtering approach is the nonlinear phase response of the compensated system which may result in significant enlarged uncertainties. The non-linearity could be eliminated by a bi-directional application of the filter, but this technique is not possible for real-time measurements. We conclude that the considered FIR compensation filter should be preferred as long as the time sample delay introduced for its construction is not critical and real-time evaluation of uncertainties is desired.

Measurement task of sensor compensation by digital filtering.

Narrow-banded sensor input signal and resulting sensor output signal.

Broad-banded sensor input signal and resulting sensor output signal.

The compensated output signals resulting from the IIR and the FIR compensation filter for the narrow-banded sensor input signal.

The compensated output signals resulting from the IIR and the FIR compensation filter for thebroad-banded sensor input signal.

Left: Frequency response of the sensor model (black) with system parameter vector (11) and the IIR compensation filter (green) designed for the available estimate (12a) of the system parameter vector for estimation of the broad-banded (25 kHz) input signal. Right: Frequency response of the actual compensated system.

Left: Frequency response of the sensor model (black) with system parameter vector (11) and the FIR compensation filter (green) designed for the available estimate (12a) of the system parameter vector for estimation of the broad-banded (25 kHz) input signal. Right: Frequency response of the actual compensated system.

Left: Uncertainty associated with the FIR compensation filter result for three different noise values obtained for the narrow-banded input. Right: Uncertainty associated with the IIR compensation filter result.

Left: Uncertainty associated with the FIR compensation filter result for three different noise values obtained for the broad-banded input. Right: Uncertainty associated with the IIR compensation filter result.