A signal can be understood either as history or as composition.

The Fourier transform is a mathematical technique that breaks a complex signal or function into the frequencies that compose it. Instead of describing how something changes over time or space, it reveals the hidden spectrum underneath. A jagged waveform can become a collection of clean sinusoidal components, each with its own frequency and strength. The transform can also work in reverse, rebuilding the original signal from those components. This change of viewpoint makes patterns, noise, periodic behavior, and bandwidth much easier to analyze.

The top equation is the one dimensional continuous form. The bottom is the two dimensional discrete form.