# The Fourier Transform

**A tutorial introduction**

## About this book

The Fourier Transform is a fundamental tool in the physical sciences, with applications in communications theory, electronics, engineering, biophysics and quantum mechanics. In this brief book, the essential mathematics required to understand and apply Fourier analysis is explained.

The tutorial style of writing, combined with over 60 diagrams, offers a visually intuitive and rigorous account of Fourier methods. Hands-on experience is provided in the form of simple examples, written in Python and MATLAB computer code. Supported by a comprehensive glossary and an annotated list of further readings, this represents an ideal introduction to the Fourier transform.

Published April 2021

Paperback ISBN: 9781916279148

Hardback ISBN: 9781916279155

Download Chapter 1 (PDF, 5.4MB)

## Contents

**Preface**

**1. Waves**

1.1 Why Fourier transforms matter

1.2 A sketch of Fourier analysis

1.3 Making waves

1.4 Changing phases

1.5 A simple demonstration

1.6 Jean-Baptiste-Joseph Fourier

**2. Mixing waves**

2.1 Setting the scene

2.2 Reconstructing functions from sinusoids

2.3 Functions with nonzero means

2.4 The Fourier transform

2.5 Approximating a function

2.6 Summary

**3. The parameters of a single sinusoid**

3.1 Introduction

3.2 Finding the amplitude

3.3 Finding sine-cosine coefficients

3.4 Finding the phase

**4. The Fourier transform**

4.1 Un-mixing sinusoids

4.2 Orthogonal basis functions

4.3 Finding Fourier coefficients

4.4 Amplitude and phase spectra

4.7 Summary

**5. Visualising complex waves**

5.1 Complex numbers

5.2 The complex plane

5.3 Euler's theorem

5.4 Visualising complex waves

5.5 Setting the phase and amplitude

**6. The complex Fourier transform**

6.1 Mixing complex waves

6.2 The complex Fourier transform

6.3 Fourier pairs

6.4 Summary

**7. Properties of Fourier transforms**

7.1 Introduction

7.2 Dirichlet conditions

7.3 Conjugate variables

7.4 The sampling theorem and aliasing

7.5 How many Fourier components?

7.6 The addition theorem

7.7 The shift theorem

7.8 Parseval's theorem

7.9 The convolution theorem

7.10 Counting Fourier parameters

7.11 Fourier transform of a Gaussian

7.12 Trading frequency for time

7.13 Information theory and Fourier transforms

7.14 The fast Fourier transform

7.15 Two-dimensional Fourier transforms

**8. Applications**

8.1 Satellite TVs, MP3 and all that

8.2 Deconvolution

8.3 Fraunhofer diffraction

8.4 Heisenberg's uncertainty principle

8.5 Crystallography

**Further reading**

**Appendices**

A. A simple example in Python

B. A simple example in MATLAB

C. Mathematical symbols

D. Key equations

E. Glossary

**Index**