Need explanation on Laplace Transform and Fourier Transform

[hilman]

Hello guys. I need an easy explanation regarding Laplace Transform and Fourier Transform. I know it is quite a mathematics question but I need an explanation in which it has something to do with engineering. I already search a bit about them but still cannot find and explanation that easy enough to be understood (like analogue and others). And also, I want to know besides s=jw (in which a Laplace Transform becomes a Fourier Transform), what kind relationship these two have?

Thanks.

 

[MarcusAgrippa]

I'm not sure what kind of answer you want. The Fourier transform expresses a function as a linear combination of the complex exponentials of the form $e^{ikx}$ or $e^{i \omega t}$. These functions are linearly independent functions that span an infinite dimensional linear (or, vector) space. The Fourier coefficients are the components of the function in this basis.

The Laplace transform enlarges the function space by allowing the k or $\omega$ to be complex numbers and does not restrict them to reals.

This probably is not the answer you want. But it is an accurate description of the math content, and any "engineering interpretation" can only alter the interpretation by assigning well defined physical correspondents to the basis elements. For example, you might interpret $e^{i\omega t}$ as a particular type of signal, and the function f(t) as some composite signal of these basic signals.

 

[Chestermiller]

I have the feeling you are asking "what are Laplace transforms used for in engineering", correct?

Chet

 

[donpacino]

Hello guys. I need an easy explanation regarding Laplace Transform and Fourier Transform. I know it is quite a mathematics question but I need an explanation in which it has something to do with engineering. I already search a bit about them but still cannot find and explanation that easy enough to be understood (like analogue and others). And also, I want to know besides s=jw (in which a Laplace Transform becomes a Fourier Transform), what kind relationship these two have?

Thanks.

essentially the laplace and Fourier transform allow you to see the frequency components of a signal.

This is useful for both the design of system, and the analysis of real systems with noise.
Think of a radio. if you want to receive one frequency, but block all others, you first need to extract that frequency data. to do that digitally, you would take the Fourier transform.

The frequency domain is used A LOT in engineering

 

[EM_Guy]

The Fourier transform transforms a function of time to a function of frequency. Any given signal can be viewed as the sum of its spectral components. That is, a signal can be viewed as the sum of sines and cosines of various amplitudes and frequencies.

The Laplace transform transforms a function of time to a function of s (s = complex frequency). To really understand the Laplace transform, you need to understand the complex frequency s. Get familiar with the concept plane and the idea of voltage or current sources that are exponentially damped sinusoids.

 

[Chestermiller]

You guys neglected to mention the important use of Laplace Transforms to solve ordinary- and partial differential equations, both single equations and sets of equations.

Chet

 

[donpacino]

You guys neglected to mention the important use of Laplace Transforms to solve ordinary- and partial differential equations, both single equations and sets of equations.

Chet

good point. specifically it allows you to solve them with algebra instead of calculus!