Fourier Series

Published May 19, 2022  -  By Marco Garosi

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):

f(t)=n=+cnej2πnTt,nZ f(t) = \sum_{n = -\infty}^{+\infty} c_n e^{j \frac{2 \pi n}{T} t}, n \in \Z

Fourier Series - Synthesis equation

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

cnC=1TT2T2f(t)ej2πnTtdt,nZ c_n \in \mathbb{C} = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t) e^{-j \frac{2 \pi n}{T} t} dt, n \in \Z

Fourier Series - Analysis equation

If cnRc_n \in \R (so if θn=0\theta_n = 0), then ej2πnTte^{j \frac{2 \pi n}{T} t} in the synthesis equation gets scaled only — it has no initial phase.

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