Mathematical modelling of Signals – Processes – Systems - Signals, Processes, and Systems - page 502

LAPLACE- transformation
Thomas BROMWICH, Karl Willy WAGNER, John R. CARSON and Gustav DOETSCH
developed the LAPLACE- transformation to a point where it was possible use it in prac-
tice. It attempted to eliminate the problem arising with the FOURIER- transformation by
a functional transformation. However, the amount of describable time functions had to be
reduced and several limit problems resolved. The demonstration of the propositions of the
LAPLACE- transformation is often mathematically “very challenging”.
Operational calculus of MIKUSINSKI
This algebraically based operational calculus was developed in the 1950s by the Polish
mathematician Jan MIKUSINSKI. It is based on HEAVISIDE´s operational calculus and
rebased it using algebraic methods in a new mathematically precise way.
Advantages of the operational calculus of MIKUSINSKI:
• an operator instantaneously represents a mathematical model of the system.
• no detour via the image domain (frequency domain) is necessary, you always work in
the original domain (time domain).
• Convergence assessment and its resulting limitations are not necessary.
• Working with distributions to describe the DIRAC- impulse (and similar signals) is
not necessary.
Disadvantages of the operational calculus of MIKUSINSKI:
• The algebraic foundation is mathematically highly abstract. Practising engineers with
limited knowledge of algebra might find it hard to visualize.
• The transition to “imaginary frequencies” is often used in practice and thus the spectral
demonstration of signals is not immediately apparent.