Instrumentation & Measurement Magazine 23-2 - 61

Inverse Algorithms-Powerful Tools
to Improve Measurement Systems
Tamás Dabóczi

easurement systems suffer from problems due to
the imperfectness of sensory devices and measuring channels, and the shortage of available
information concerning the investigated phenomena itself.
This imperfectness and shortage result in unwanted alterations and perturbations of the measured values and/or signals.
Better results can be achieved if these alterations and perturbations are counteracted. The role of the inverse algorithms is this
counteraction in order to improve the quality of measurement.
This paper deals with the challenging issue of addressing
various complexity levels of inverse problems and provides solutions illustrated via different application examples.
Throughout this paper, the following terminology is used:
Quality of measurement is expressed on one hand by its accuracy, i.e., how close is the measured value to the true (or
theoretical) value, while on the other by its precision, i.e., how
close the measurements are to each other [1].
Distortion is the unwanted deterministic alteration of the
original shape (or other characteristic) of the measured signal. Addition of noise or other outside signal (interference)
is not considered distortion. If the measured signal can be expressed as an input-output relation by y(t) = F(x(t)) and the
inverse function F−1 can be found, then the distortion can be
eliminated by intentionally (anti)distorting the output. (Sometimes, the inverse can be more easily expressed in a transform
(e.g., frequency) domain.) If the input-output relation can only
be approximated, e.g., by using system identification methods, and its inverse exists, then an approximate elimination
is possible.
Disturbance is the unwanted perturbation of the original
shape (or other characteristic) of the measured signal. Typically, added noise and other not identified or not identifiable
outside signals (interferences) are considered disturbance. The
disturbance cannot be eliminated, only reduced.
Inverse algorithms are methods that struggle against the
deterministic distortion, taking into account that it is an

unavoidably simultaneous fight against the distortions and
disturbances to achieve optimum signal reconstruction at the
output of the measurement system. Balancing this simultaneous counteraction is a rather sensitive issue since the solved
problem is an inverse problem, which is typically ill-posed,
since arbitrarily small errors in the measurement data may
lead to indefinitely large errors in the solutions. If the disturbances are not properly suppressed, then the precision of the
measurement will be questionable, since the variance of the
measurement will be unacceptably high, while if the distortion
is not compensated enough, then the measurement accuracy
will be poor.
Throughout this paper, it is assumed that distortion of the
measurement system is a priori known. If this knowledge is
the result of previous system identification, it is important to
emphasize that its quality is crucial: the inverse algorithms
introduced in this paper cannot compensate mismodeling.
Moreover, system identification influences also the degree
of ill-posedness of the inverse problem. As an illustration,
let us assume that the input-output relationship is given by
a transfer function in the frequency domain. If the system
identification step determines a roll off rate larger than it is in
reality, then it worsens the condition number of the inverse.
It has been shown in [2] that these two steps influence each
other, and if in doubt, system identification needs to be rather
under- than over-regularized from the point of view of signal
reconstruction. Combined system identification and signal reconstruction might result in better quality measurements, but
this type of optimization is poorly investigated and is out of
the scope of this paper.
The paper is organized as follows: First, the ill-posedness
will be demonstrated on an inverse filtering problem, and the
concept of regularization will be shown. Next, different complexity levels will be introduced and supported by application
examples. This gives insight into the information processing
power of the inverse algorithms.

This work was partially supported by BME-Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC) and
National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program).
April 2020

IEEE Instrumentation & Measurement Magazine 61
1094-6969/20/$25.00©2020IEEE

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