Definition of the Fourier Series

 

1. Introduction

Most of the phenomena studied in the domain of Engineering and Science are periodic in nature. For instance, current and voltage in an alternating current circuit. These periodic functions could be analyzed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis.
A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series.
Fourier series is used to describe a periodic signal in terms of cosine and sine waves. In other words, it allows us to model any arbitrary periodic signal with a combination of sines and cosines. 

 

 

Fig: Sine Wave

 

 

 

Fig: Triangular Wave 

 

Fig: Sawtooth Wave 

 

 

Fig: Square Wave

 

2. The common form of the Fourier series

Sinusoidal functions are periodic over 2π angular distance.
For a periodic function f(x), (let’s assume a function other than sinusoidal function i.e., a square wave with a period of 2π) 

 

 

 

** I’ve often used the terms ‘cos(nx)’ and ‘sin(nx)’ throughout the article. They actually indicates that cos(nx) and sin(nx) are periodic on the interval 2π for any integer n
e.g., for n = 1, cosx has a period of 2π/1 = 2π
        for n = 2, cos2x has a period of 2π/2 = π and so on.
The signal's direction of propagation is indicated by the letter ‘x’. It denotes the signal is propagating along x-axis. This axis can also be used as a ‘time’ axis.

We know that period of a sinusoidal signal is 2π/T. T is the value after which the signal repeats.
In the above equations, we assumed that the periodic signal f(x) has a period of 2π. So now, if a function is periodic on the interval [-L, L], we'll talk about how to expand it into a Fourier series.