Operator calculation

In electrical engineering and system theory of communications engineering, operator calculation is understood to mean various historically grown mathematical calculi for describing the behavior of linear time-invariant systems . Instead of the “classic” description using differential equations and differential equation systems and their complex solution, the operator calculation describes the behavior of the elementary components and complex systems using operators and thus leads the differential equations back to algebraic equations .

Mathematically, there is a function vector space finite in the dimensions , which can always also be formulated explicitly algebraically.

A system is described by the following simple algebraic relationship:

{\ displaystyle \ mathrm {effect = system characteristic \ star cause}}

In all operator calculations the difference between the signals and the system characteristics disappears. Both are represented equally by the respective operators.

The different operator calculations were created in the historical order given below:

The complex AC bill

This symbolic method of alternating current calculation introduces the complex resistance operator (and others) (as the so-called “jω calculation”) , but is bound to stationary sinusoidal signals. The introduction of the complex frequency in the extended symbolic method cannot in principle change anything.

The Heaviside calculus

Oliver Heaviside extended the symbolic method of alternating current calculation empirically for arbitrary signals by introducing the differential operator and using it like a “normal” variable. However, this Heaviside operator calculation sometimes led to incorrect results in the (“somewhat difficult”) interpretation (i.e. under conditions that were not specifically to be specified) and was not precisely justified mathematically. ${\ displaystyle p = {\ frac {d} {dt}}}$

The HY calculus is an extension and generalization of the Heaviside calculus .

The Laplace Transform

The Laplace transformation worked out by Thomas John l'Anson Bromwich , Karl Willy Wagner , John Renshaw Carson and Gustav Doetsch in a practical manner tried to eliminate these problems (based on the Fourier transformation ) by means of a functional transformation . For this, however, the number of writable time functions had to be restricted and various limit value problems had to be solved to justify them. The demonstration of the theorems of the Laplace transformation is often mathematically "very demanding".

The operator calculation according to Mikusiński

This algebraically based operator calculation was developed in the 1950s by the Polish mathematician Jan Mikusiński . It is based on Heaviside's operator calculation and justifies it with exact mathematical re- establishment using algebraic methods .

Advantages of the operator calculation according to Mikusiński

An operator is directly a mathematical model of the system.
No detour via an image area (frequency area) is necessary, but you always work in the original area (time area).
Convergence studies and the resulting restrictions are not necessary.
It is not necessary to work with distributions to describe the Dirac impulse (and similar signals).

Disadvantages of the operator calculation according to Mikusiński

The algebraic justification is mathematically very abstract and unreasonable for "practicing engineers" with little algebraic training.
The transition to the “imaginary frequency” that is often used in practice and thus the spectral representation of signals is not immediately obvious.

For this reason and because of the extensive literature, the Laplace transformation is still the most widely used method of operator calculation in engineering practice as well as in teaching .

literature

Jan Mikusiński: Operator calculation . German Science Publishers, Berlin 1957.
FH Lange: Signals and Systems . tape 1 : Spectral representation. Verlag Technik, Berlin 1965.
Gerhard Wunsch : History of Systems Theory . Akademie-Verlag, Leipzig 1985.

Individual evidence

↑ Wolfgang Mathis: Theory of nonlinear networks . Springer-Verlag, 1987, ISBN 3-540-18365-5 .

[1] Wolfgang Mathis: Theory of nonlinear networks . Springer-Verlag, 1987, ISBN 3-540-18365-5 .