﻿ Operator theory

# Operator theory

Under operational calculus is understood in electrical engineering and systems theory in communications engineering various historically developed mathematical calculi for describing the behavior of linear time-invariant systems. Instead of the "classic" description by differential equations and systems of differential equations and their complex solution, the operational calculus describes the behavior of the elementary components and the complex systems by operators and thus leads the differential equations to algebraic equations back.

Mathematics lies in the dimensions of finite function vector space in front, which can always be explicitly formulated algebraically.

A system is described by the following simple algebraic relation:

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

The different Operators bills originated in the historical order given below:

## The complex AC circuit analysis

This symbolic method of alternating current calculation results ( as so-called " j.omega calculation" ) is a complex resistance operator ( and others), but is bound to stationary sinusoidal signals. The introduction of the complex frequency in the extended symbolic method can make no difference in principle.

## The Heaviside calculus

Oliver Heaviside extended the symbolic method of alternating current account empirically for arbitrary signals by introducing the differential operator and as a used him "normal" variable. These Heaviside operational calculus resulted in the ( " somewhat difficult" ) Interpretation sometimes (ie not specific to specifying conditions ) to erroneous results and was not exactly mathematically justified.

An extension and generalization of Heaviside calculus presents the HY- calculus dar.

## The Laplace transform

The Thomas John I'Anson Bromwich, Karl Willy Wagner, John Renshaw Carson and Gustav Doetsch practicable elaborated Laplace transform tried these problems ( starting from the Fourier transformation ) to eliminate by a functional transformation. For this purpose, but had the amount of the recordable time limited functions and are solved in support of various limit problems. The proof of the sentences of the Laplace transform is often mathematically " very challenging ".

## The operational calculus by Mikusiński

This algebraic founded operational calculus was developed in the 50s by Polish mathematicians January Mikusiński. It is based on the Heaviside operational calculus and the justification for it mathematically exact new with algebraic methods.

Benefits of operational calculus by Mikusiński

• An operator is immediately followed by a mathematical model of the system.
• There is no detour over an image area (frequency range ) is necessary, but it always works in the original domain ( time domain ).
• Convergence studies and the resulting restrictions are not necessary.
• Working with distributions to describe the Dirac pulse ( and similar signals) is not necessary.

Disadvantages of the operational calculus by Mikusiński

• The algebraic reasoning is mathematically very abstract and trained for little algebraically " practicing engineers " not vivid.
• The transition to practice often used " imaginary frequency " and thus the spectral representation of signals is not immediately obvious.

Therefore, and due to the extensive literature both in the practice of engineering activities as well as in teaching today, the Laplace transform, the method of operational calculus usually applied.

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