What is Laplace Transform?

Answer 1

The two main techniques in signal processing, convolution and Fourier analysis, teach that a linear system can be completely understood from its impulse or frequency response. This is a very generalized approach, since the impulse and frequency responses can be of nearly any shape
or form. In fact, it is too general for many applications in science and engineering. Many of the parameters in our universe interact through differential equations. For example, the voltage across an inductor is proportional to the derivative of the current through the device. Likewise,
the force applied to a mass is proportional to the derivative of its velocity. Physics is filled with these kinds of relations. The frequency and impulse responses of these systems cannot be arbitrary, but must be consistent with the solution of these differential equations. This means that their impulse responses can only consist of exponentials and sinusoids. The Laplace transform is a technique for analyzing these special systems when the signals are continuous.

Answer 2

The Laplace Transform of a function f(t) is defined for all real numbers as:

F(s)= integral from 0 to infinity of e^(-st) f(t) dt.

This integral transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, with s. This changes integral equations and differential equations to polynomial equations, which are much easier to solve.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory.

from Wikipedia ( http://en.wikipedia.org/wiki/Laplace_transform )

Answer 3

In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

The Laplace transform is an important concept from the branch of mathematics called functional analysis.

In actual physical systems the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system.

The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory.

The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.

Answer 4

In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

The Laplace transform is an important concept from the branch of mathematics called functional analysis.

In actual physical systems the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system.

The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory.

The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.

Answer 5

The Laplace remodel is a mathematical technique used to transform a actual-valued function right into a complicated valued one. It has purposes in many, mnay aspects of physics and engineering including optics, administration thought, sign processing and electric powered engineering. For a function f(t) the Laplace remodel is F(s) = indispensable from 0* to infinity of exp(-s t) f(t) dt as an occasion, the Laplace remodel of sin(t) is one million/(one million+s²) s is many times a complicated extensive form. the explanation that is functional is as a results of the fact integration and differentiation substitute into multiplication and branch interior the Laplace area. This very oftentimes simplifies the prognosis of actual or mathematical structures. * (the decrease shrink isn't strictly 0 yet extremely the shrink as an aribitrarily samll quantity has a tendency to 0).

Answer 6

Laplace transform is Linear time-invariant systems, Like electrical circuits, Harmonic oscillators and Optical devices.

Answer 7

in a nutshell .

it is a special transformation of a function f(x) to a function g(s), wich involves an integral.

calculations can be done with the resulting g(s).
sometimes these calculations are easier todo than with the original f(x).

finally the g(s) can be transformed back .

google for this.