Fourier Series

What is a Fourier series?

A Fourier series allows you to decompose a periodic function into a sum of complex sinusoidal waves of frequencies that is integer multiple of its fundamental frequency.

Introduction

We can express any closed function as a vector sum of rotating circles moving at different speeds, initial angle and amplitude.

Note

For a function $e^{n \cdot 2 i π t}$ . It is rotating $n$ times about the origin within a span of $1 s$

So any function can be expressed as $f (t)$ , $f : R \mapsto C$ . Where $f$ is a linear combination of rotating complex numbers in the form of $r e^{i t}$ where $t \in [0, 1]$

f (t) = \dots + c_{- 1} e^{- 1 \cdot 2 π i t} + c_{0} e^{0 \cdot 2 π i t} + c_{1} e^{1 \cdot 2 π i t} \dots

or more simply

f (t) = \sum_{n = - \infty}^{\infty} c_{n} e^{n \cdot 2 π i t}

In addition, since

\int e^{a x} d x = \frac{1}{a} e^{a x}, a \neq 0

thus

\int_{0}^{1} e^{n 2 π i t} d t = \frac{1}{2 π i n} {[e^{2 n π i t}]}_{0}^{1} = \frac{1}{2 π i n} (1 - 1) = 0, n \neq 0

However, if $n = 0$ .

\int_{0}^{1} e^{0 \cdot 2 π i t} d t = 1

To find $c_{0}$ we can take the integral of $f (t)$ from $0$ to $1$ .

\begin{aligned} c_{0} & = \int_{0}^{1} f (t) d t \\ = \dots + c_{- 1} \int_{0}^{1} e^{- 1 \cdot 2 π i t} d t + c_{0} \int_{0}^{1} e^{0 \cdot 2 π i t} d t + c_{1} \int_{0}^{1} e^{1 \cdot 2 π i t} d t + \dots \\ = \dots 0 + 0 + c_{0} + 0 + 0 \dots \end{aligned}

In addition, to find $c_{1}$ , we can take the integral of $f (t) \cdot e^{- 1 \cdot 2 π i t}$ from $0$ to $1$ .

\begin{aligned} c_{1} & = \int_{0}^{1} f (t) \cdot e^{- 1 \cdot 2 π i t} d t \\ = \dots + c_{- 1} \int_{0}^{1} e^{(- 1 - 1) \cdot 2 π i t} d t + c_{0} \int_{0}^{1} e^{(0 - 1) \cdot 2 π i t} d t + c_{1} \int_{0}^{1} e^{(1 - 1) \cdot 2 π i t} d t + \dots \\ = c_{1} \end{aligned}

Thus, we essentially make the integral in the $c_{1}$ expression equate to 1. More generally

c_{n} = \int_{0}^{1} f (t) e^{- n \cdot 2 π i t} d t