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Disadvantages of the Laplace Transformation Method

That is, you can only use this method to solve differential equations WITH known constants. If you do have an equation without the known constants, then this method is useless and you will have to find another method.Oct 22, 2020

Laplace Transform is widely used by electronic engineers to solve quickly differential equations occurring in the analysis of electronic circuits. 2. ... Laplace Transform is used to simplify calculations in system modeling, where large number of differential equations are used.

A Fourier transform is a special case of a Laplace transform. That is, the Laplace transform is a generalization to the Fourier transform to include damping, or signals that decay with time.

4.3. The Laplace transform. It is a linear transformation which takes x to a new, in general, complex variable s. It is used to convert differential equations into purely algebraic equations.

DTFT: Discrete Time... hence time signal is in samples, Fourier Transform... hence Fourier transform is a continuous function. That is why time signal is represented as x[n], n being discrete sample nos.

• First of all, note that when we talk about the Laplace transform, we very often mean the unilateral Laplace transform, where the transformation integrals starts at \\$t=0\\$ (and not at \\$t=-\\infty\\$), i.e. with the Laplace transform we usually analyze causal signals and systems. With the Fourier transform this is not always the case.

• However, if you want to understand what a system does when you flip a light switch, you typically need Laplace transforms. In addition, the Fourier Transform exists along the j ω axis of the two sided Laplace Transform so Fourier is a slice of Laplace. The σ axis captures the dissipation/regeneration properties of Linear Systems.

• It's not true that the Fourier transform can only capture the steady state behavior. The frequency response of an LTI system, which is the Fourier transform of its impulse response, completely characterizes its behavior, and hence it covers transients as well as the steady state response.

• The inverse Laplace transform of a function F ( s) is taken to be the function f ( t) in such a way that L { f ( t )} = F ( s ), and in the usual mathematical notation we write, L -1 { F ( s )} = f ( t ). The inverse transform can be made unique if null functions are not allowed.

First of all, note that when we talk about the Laplace transform, we very often mean the unilateral Laplace transform, where the transformation integrals starts at \$t=0\$ (and not at \$t=-\infty\$), i.e. with the Laplace transform we usually analyze causal signals and systems. With the Fourier transform this is not always the case.

However, if you want to understand what a system does when you flip a light switch, you typically need Laplace transforms. In addition, the Fourier Transform exists along the j ω axis of the two sided Laplace Transform so Fourier is a slice of Laplace. The σ axis captures the dissipation/regeneration properties of Linear Systems.

It's not true that the Fourier transform can only capture the steady state behavior. The frequency response of an LTI system, which is the Fourier transform of its impulse response, completely characterizes its behavior, and hence it covers transients as well as the steady state response.

We can easily see that what is written in the Fourier transform as exp (-jwt), now becomes exp (-st) where s = sigma + jw. The sigma is the real or called as the exponential part, where the jw is the imaginary or called sinusoidal part.