Instrumentation & Measurement Magazine 23-2 - 66

Fig. 4. Reconstruction in the case when the physical quantity can be directly measured by a sensor.

Fourier spectrum of the
actual noise record is important, not its expected
value based on infinite
trials.) Based on our experience, more than three
parameters are challenging
to automatically handle,
since the error surface gets
more complex, having local
minima. The parameters
of the inverse filter are:
λ = 0.0013, γ = 0.51, δ = 1.9.

Extension of
Possibilities Beyond
Actual Technological
Limits
There are applications
requiring accurate and precise measurements even if
they are costly. Among others, calibration laboratories
belong to this group. Such
laboratories need calibration instruments to provide
higher accuracy and precision than the instrument
they calibrate. There are
cases when the technology
reaches its borders, and the
bandwidth or accuracy of
Fig. 6. Investigation of insulations by non-destructive high voltage lightning impulse measurements. (Measurements were
the calibration instrument
accomplished at the High Voltage Laboratory at Swiss Federal Institute of Technology, ETH Zürich.)
is only slightly better than
that of the most recent in(Fig. 6). The experiment is accomplished on a reduced signal struments. In this case, inverse algorithms can help again to
level, allowing the use of resistive divider. Divider Under Test numerically compensate for the distortions that we could not
measures the fast pulse with large distortion, both in terms of eliminate in the measurement hardware.
signal shape and signal parameters (e.g., ∼30% peak error).
However, inverse filtering can improve the measured waveform comparable to that of the reference divider (Fig. 7). Here,
we used Tikhonov type regularization with three free parameters, having the following form for the inverse filter:
Fig. 5. Chain of signal processing: from analog physical quantity to digital signal and digital post processing.

	

K( f ) =

H( f )

∗

H ( f ) + λ + γ L( f ) + δ L( f )
2

2

4

.	(14)

The additional two regularization operators-compared
to (7)-provide more freedom to match the required noise reduction at different frequency bands. Theoretically, infinite
number of operators would be ideal; however, in practice we
do not have enough a priori knowledge about the energy relations of the signal to be reconstructed and the noise record
at different frequency bands. (Here, the reconstruction needs
to be accomplished from one single measurement, thus the
66	

Fig. 7. Bandwidth enhancement by means of inverse filtering.

IEEE Instrumentation & Measurement Magazine	

April 2020



Instrumentation & Measurement Magazine 23-2

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