# Fourier series

## Introduction

Born in Auxerre, France in 1768, Jean Baptiste Joseph Fourier was orphaned at the age of eight. Later he was instructed by Benedictine monks who taught at and ran a military college.

Years later (1822) Fourier's genius became evident when he discovered that he could represent a periodic function as a sum of sinusoids. It may be interesting to note that what has come to be known as the Fourier series was invented while Fourier was studying heat flow.

Even though Fourier had discovered a powerful tool, his peers were slow in accepting it. This may be because he provided to rigorous proof to show that his series was an accurate representation of a periodic function. Later, P.G.L. Dirichlet was able to present an acceptable proof.

Information used in the introduction has been adapted from Linear Circuit Analysis by DeCarlo & Lin and Fundamentals of Electric Circuits by Alexander and Sadiku.

## Periodic Functions

A continuous time signal $x(t)$ is said to be periodic if there is a positive nonzero value of T such that

$s(t + T) = s(t)$ for all $t$

## Dirichlet Conditions

The conditions for a periodic function $f$ with period 2L to have a convergent Fourier series are as follows:

### Theorem:

Let $f$ be a piecewise regular real-valued function defined on some interval [-L,L], such that $f$ has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to $f$ when $f$ is continuous and to the arithmetic mean of the left-handed and right-handed limit of $f$ at a point where it is discontinuous.

## The Fourier Series

A Fourier series is an expansion of a periodic function $f$ in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. One way to represent a Fourier series is

$f(t) = \sum_{k= -\infty}^ \infty \alpha_k e^ \frac{j 2 \pi k t}{T}$.