**Fourier Series** is a mathematical tool, named after French mathematician Joseph Fourier, that allows us to decompose a *periodic* function as a **sum of sine and cosines waves**. Each of those sines/cosines has a frequency which is an integer multiple of the periodic function’s fundamental frequency.
Fourier Series is a fundamental part of signal processing and will later be extended with the Fourier Transform, which extends it to non-periodic signals.

This post is part of a series on Image and Signal Processing. If you are looking for the Fourier Transform, you may read the related note.

## Mathematical definition

A Fourier Series is a sum of sine waves and is so defined (** synthesis equation**):

Every “iteration” has to be provided a
$c_n$ value.
$c_n$ values can be computed with the so-called ** analytis equation**:

If $c_n \in \R$ (so if $\theta_n = 0$), then $e^{j \frac{2 \pi n}{T} t}$ in the synthesis equation gets scaled only — it has no initial phase.