Introduction

Welcome to the wonderful world of Fourier Analysis! Despite looking difficult at first glance, Fourier Analysis is actually pretty easy and in the next few tutorials we’ll be running through two very common types of Fourier Analysis to help you completely understand it. There are four types in total and several other types of transforms we’ll be looking at, but we won’t be focussing on these until much further on (read the introduction for more information).

What is Fourier Analysis?

So what actually is Fourier Analysis? Simply put:

Fourier Analysis is the process of approximating a function in terms of nothing but simple trigonometric functions.

But what does that actually mean? As you may (or may not) know, the easiest type of signal to create is a sine wave or cosine wave. If we take a signal (a sound bite of speech for example), it can be very difficult to define that signal as a mathematical function, so instead we break that down into a sum of sine and cosine waves (our trigonometric functions) using Fourier Analysis.

Let’s take a simple example, how can we define a square wave as a mathematical function? Well unlike a sine or cosine wave, we don’t have the option of simply defining it as:


Instead it is possible to create a square wave by adding together multiple sine waves:

Let’s visualise this:

Creating a Square Wave

Creating a Square Wave

As you can see once all three sine waves have been added together, it resembles a square wave. If we add more sine waves:

We can start to make it look more and more like a square wave:

Square Wave

Square Wave

We can continue to add more and more sine waves and the wave will become almost exactly like a square wave (although it can never truly perfect).

In this manner, it is possible to create any wave simply by adding sine/cosine waves together, as long as the coefficients and multiples of x are correct (which we calculate using Fourier Analysis).

Types of Fourier Analysis

So before we dive in, we’ll first look at the four main categories of Fourier Analysis:

  • Periodic – Continuous: Fourier Series
  • Aperiodic – Continuous: Fourier Transform
  • Periodic – Discrete: Discrete Fourier Transform
  • Aperiodic – Discrete: Discrete Time Fourier Transform

Or:

Types of Fourier Analysis

Types of Fourier Analysis

We select which type of Fourier Analysis we will use based on the type of signal we are dealing with. In the square wave example above we would have used Fourier Series, since the signal we were trying to reproduce was a periodic, continuous function.