The Fourier Transform assigns a function f ( t ) in the original (time) domain with a variable t to a function F ( \omega ), where \omega describes the frequency domain and \mathcal{F} is the operator of the transformation.
| \stackrel{\text{Original function}}{f(t)} \qquad \stackrel{\text{FT}}{\circ - \bullet} \qquad \stackrel{\text{Transformed function}}{F(\omega) = \mathcal{F[f(t)]}} |
Fourier transformations can be used to transform differential equations into algebraic equations, as a derivative turns into a multiplication with \omega in the transformed domain. Hence, the algebraic equation can be solved easily in the transformed domain and the original solution can be found be the inverse transformation back to the original domain. Furthermore, the Fourier Transform can be used to perform investigations in the transformed domain where the physical relations are more obvious.
Theory
The transformation rule is given by
| \begin{alignat*}{2} F(\omega) &= \int_{-\infty}^\infty f(t) e^{-i\omega t} \,dt \qquad &\text{Fourier Integral}\\ f(t) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} \,d\omega \qquad &\text{Inversion Formula} \end{alignat*} |
Note, that the Fourier transformation is just the limiting case of the Fourier series of a signal with an infinite period T \rightarrow \infty.