What is the Fourier transform & Laplace transform used for?

Question

I want some real life applicatiuons of it.

Answer 1

Almost everything the deals with waves is described using Fourier transforms: radio transmissions, sounds waves, optics, etc.

Laplace transforms are used in circuit equations that have on/off switches. The switch is described as a step-function, and Laplace transforms can easily describe this behavior.

These are just examples. Both transforms are extremely useful. They basically convert differential equations into algebraic equations and make life easier.

Answer 2

In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. A useful analogy is the relationship between a series of pure notes (the frequency components) and a musical chord (the function itself). In mathematical physics, the Fourier transform of a signal can be thought of as that signal in the "frequency domain." This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function.
In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. 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.

http://en.wikipedia.org/wiki/Fourier_transform
http://en.wikipedia.org/wiki/Laplace_transform

Answer 3

The best example of a real life example is electronics. They are used to design filters. The fourier transforms relates time based systems to frequency based systems.

The classic differential equations example uses a suspension in a car and how it responds to the road. The system is designed using a laplace function and shown how it will react using a step function (step response)