Practical significance of laplace trasformation
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A particular notion of study of Signal systems lies not just in studying the time domain analysis that is how the signal behaves over a particular time interval but also what components of signals (a.k.a frequency components with particular amplitude) lies in signals i.e frequency domain analysis. Frequency analysis lets us study which frequency components are to be added or whose magnitude should be changed to bring the system at particular frequency response (natural frequency of system) or phase
When we study how a particular output signal f(x) will behave with system transfer function and input signal h(x) (impulse, step or ramp function), we use Convolution . But convolution is hard to be done, by virtue thanks to almighty we have basic theorem that Convolution of 2 signals in time domain is equal to multiplication of signals in frequency domain. And the tool for Conversion is Fourier Transform
But since in a practical world nothing is ideal and Fourier series only gives ideal responses in frequency domains, we need a damping factor , and this is given by Laplace Transform. Thus Fourier contains only imaginary terms (Sinusoidal), but Laplace is real (Exponential)+Imaginary (Fourier term)
Another feature of Laplace is that it converts non linear differential equations, sometimes non homogeneous to linear forms. Think of series RLC circuits, we have differentials, integrals and linear terms all together in single equation. This becomes might be easy but complex circuits become tough to solve but Laplace Equation does it magic making process very simple
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Luciano Zoso
Luciano Zoso, digital signal processing engineer, 35 years of experience
Answered Sep 15, 2016 · Author has 922 answers and 1.5m answer views
The Laplace transform is an essential part of the mathematical background required of engineers, physicists, mathematicians and other scientists. This is because it provides easy and effective means for the solution of many problems arising in various fields of science and engineering.
The Laplace transform is a frequency-domain approach for continuous time signals. It converts a function of a positive real variable t (usually time) to a complex function of a complex variable s (frequency).
The Laplace transform is particularly useful in solving linear ordinary differential equations such as those encountered in the analysis of electronic circuits. Since differentiation and integration become multiplication and division by s, the transform turns integral equations and differential equations to polynomial equations, which are much easier to solve.