Instrumentation & Measurement Magazine 23-2 - 64

may want to approximate nonlinearity with the best linear dynamic model [7].
Static nonlinearity can be described by a static transfer
function. Its inversion is quite straightforward. The inverse
can be either explicitly expressed or approximated by finite elements of a Taylor series or piecewise linear approximation.
The inversion gets challenging if the noise level of the measurement is not negligible, and there are saturating parts in the
static nonlinearity. At saturating parts, the inverse amplifies
extremely the noise, producing a signal-level dependent noise
amplification (Fig. 3). We faced this problem by trying to improve the sound quality of optically recorded old movies that
suffer from nonlinear distortions.
To overcome on this problem, we can introduce some kind
of regularization to the nonlinear inversion, similar to that of
inverse filtering (see (7)). If the inverse of the static nonlinearity is denoted by K(y), first let us approximate it by the first two
elements of its Taylor series:
	

K ( y0 + Δy ) ≈ K ( y0 ) +

dK ( y )
dy

⋅ Δy.	(12)
y = y0

Now, instead of the exact derivative, a regularized one is
substituted [8]:

	

dy

→
y = y0

dx


 dN ( x )
 dx



x= xˆ 0

2


 +λ


x= xˆ 0 

,	(13)

where N(x) stands for the static transfer function of the nonlinearity. The regularized inverse is obtained by numerical
integration of the above derivative. The offset can be determined from other boundary conditions. The reconstruction
with regularized inverse can be seen in Fig. 3d.

Complexity of Inverse Algorithm
The whole measurement system consists of many different
parts (Fig. 4). The sensor first transforms the energy of the
physical quantity to be observed to a form that can be measured electronically and provide an analog electrical output
(voltage, current, resistance, capacitance etc.). Analog signal
conditioning transforms this output to a voltage appropriate
in level and range for the AD converter and might perform
impedance matching, galvanic decoupling, overvoltage protection, noise filtering, antialias filtering, etc. This part is called
many times analog signal processing (ASP). AD converter
transforms the analog signal into digital domain (sampling
and quantization), and finally, a digital processor (microcontroller, general purpose processor or digital signal processor)
calculates the computation that is worth accomplishing in
digital domain. Both distortion and disturbance can affect all
parts of the whole signal chain; nevertheless, the measurement system can be considered also as one system and model
all distortions and disturbances combined, as we will do in the
64	

Case 1: Physical Quantity Can be
Directly Measured by a Sensor
Let us start with the case when there is an appropriate sensor
for the physical quantity to be observed, and there is no other
mechanical or physical (temperature/radiation, etc.) restriction that would restrict its use (Fig. 5). Inverse algorithm or
reconstruction has the duty to compensate for the distortion,
suppress the noise, and find the good balance between precision and accuracy of the estimate in ill-posed cases. We will
show application examples with different motivating reasons.
The first one tries to reduce the price of the measurement system, as digital compensation of errors might be less expensive
than improving the physical measurement system. The second
one extends technical limits, while the third one corrects errors
of non-repeatable measurements.

Affordable Measurement System with Good
Performance

dN ( x )

dK ( y )

remaining part of this paper. The complexity of the inverse algorithm depends on several factors. The first question is if the
physical quantity to be observed can be directly measured
with a sensor, or only its secondary effect can be observed. In
the latter case, the second question is if the correspondence between the physical quantity to be observed and its secondary
effect is influenced by other "disturbing" physical quantities.

Improvement of the technical properties of the sensor or the
whole measurement instrument by enhancing the hardware
is often possible but not economical. Technology can provide
a sensor with higher bandwidth, or less nonlinear distortion, but not in the price range our application requires. We
can construct advanced measurement circuits, able to compensate many distortions and suppress many disturbances,
but again, at an unreasonable cost. Numerical correction of
errors (inverse algorithms) provide the alternative. The processor required for that is either already part of the system or
is less expensive than the improvement of the hardware of the
measurement instrument. Cost effectiveness is important in
general and has a great importance for products manufactured
in large volume.
We will demonstrate the possibility of inverse filtering by
extending the bandwidth of a high voltage divider close to the
possibilities of a calibration instrument. Insulations are tested
by applying high voltage impulse (in the range of 1 MV peak)
to them and measuring the waveform. The voltage level is
gradually increased. Any defect in the insulation changes the
waveform, thus, the defect can be observed in an early stage,
without destroying the insulation (non-destructive test). The
extreme voltage level needs to be reduced by high voltage divider to the range that a traditional signal acquisition system
can accept. Our measurement setup consists of a narrow bandwidth damped capacitive divider (affordably priced), that
is unable to accurately measure short pulses (so called front
chopped lightning impulses). This is our Device Under Test
(DUT). A high bandwidth resistive divider provides the reference measurement that is designed for calibration purposes

IEEE Instrumentation & Measurement Magazine	

April 2020
Instrumentation & Measurement Magazine 23-2

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