The Fourier Transform

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