# Relationship between Fourier and Lpalace transforms

by cocopops12
Tags: fourier, lpalace, relationship, transforms
 P: 2,251 that statement doesn't state it well. but the end result makes sense to me. here is what you do: suppose you have a Linear Time-Invariant system (LTI). then the impulse response, $h(t)$ fully defines the input/output characteristic of the LTI. if you know the impulse response, you know how the LTI will respond to any input. anyway, the double-sided Laplace transform of $h(t)$ is $H(s)$. if you drive the input of that LTI with $$x(t) = e^{j \omega t}$$ then the output of the LTI system is $$y(t) = H(j \omega) e^{j \omega t}$$ same $H(s)$, just substitute $s = j \omega$. it's easy to prove, if you can do integrals.
 P: 2,251 oh, and what's easier to prove is that the Fourier transform is the same as the double-sided Laplace transform with the substitution $s = j \omega$. that's just using the definition.
 P: 43 A signal has its Fourier transform if and only if its ROC of Laplace transform contains the imaginary axis s=jw. The statement that you give is valid only for the right-hand sided signals for which the ROC is the right hand side of the poles. Fourier transform and Laplace transfrom (whether one-sided or two-sided) are not equivalent. Fourier transform can be considered as a special case of Laplace transform, that is, just set $\sigma = 0$.