The Fourier Transform: A Tutorial Introduction

by James V Stone

Overview

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.

Product Details

ISBN-13:	9781916279155
Publisher:	Tutorial Introductions
Publication date:	05/01/2021
Pages:	104
Product dimensions:	6.00(w) x 9.00(h) x 0.31(d)

Table of 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 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 The Fourier Integral

4.5 Amplitude and Phase Spectra

4.6 Conjugate Variables

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 The Sampling Theorem and Aliasing

7.4 How Many Fourier Components?

7.5 The Addition Theorem

7.6 The Shift Theorem

7.7 Parseval's Theorem

7.8 The Convolution Theorem

7.9 Logan's 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

Editorial Reviews

"This book makes an excellent supplement to amathematical methods course. In particular, Fourier series and transforms aremotivated in terms of waves and signals, and the book develops intuition froman applied perspective which I think would very much appeal to engineeringstudents. The figures are helpful throughout and nicely complement thedevelopment of the content." -Dr FRoukema, Sheffield University, England

From the Publisher