I’m not the most qualified to explain how it works or do justice to its true versatility and utility, so I suggest watching this video first as an introduction to the matter if you want to delve into the mathematics of it.

**PLEASE VIEW THIS ON A COMPUTER**

Below is a javascript program that I wrote to show how the transform can be used to interpret any signal as its input. Think of each circle to be rotating along the circumference of the larger one and smallest one has a pen that’s tracing out the path. The program can then draw whatever it is given, only changing the radius and number of the circles by breaking down the signal using a discrete Fourier transform.

To use the program just hold down your mouse over the black area and draw whatever you want. Release the mouse when you finish drawing. (Once you draw something you cannot redraw or add anything so refresh the program to make a new drawing).

Note : Instead of using a single complex input, I chose to use two different sets of epicycles to show what is going on, but the overall concept is still the same, with one set representing the real values and one representing the complex values.

See Also

But what is a Fourier series? From heat flow to drawing with circles | DE4 | 3 Blue 1 Brown (https://www.youtube.com/watch?v=r6sGWTCMz2k)