Fast Fourier Transforms

James S. Walker

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9780849371639
• eBook ISBN: 9781351448871
• Topic: Matematica
• Subtopic: Matematica applicata

This new edition of an indispensable text provides a clear treatment of Fourier Series, Fourier Transforms, and FFTs. The unique software, included with the book and newly updated for this edition, allows the reader to generate, firsthand, images of all aspects of Fourier analysis described in the text. Topics covered include :

1

Basic Aspects of Fourier Series

This chapter is a summary of the basic theory of Fourier series. Some of the deeper theorems are only quoted; their proofs can be found in [Wa], listed in this book’s bibliography. Besides their importance in applications, Fourier series provide a foundation for understanding the FFT.

1.1 Definition of Fourier Series

To understand the definition of Fourier series we will begin with the essential idea: representing a wave form in terms of frequency as opposed to time. We will denote time by x rather than t.

Suppose our wave form is described by 4 cos 2π vx, which has frequency v. Using Euler’s identity

$e^{i\varphi }=cos\varphi +isin\varphi$

(1.1)

we can write 4 cos 2π vx in complex exponential form

$4cos2\pi \nu x=2e^{i2\pi \nu x}+2e^{-i2\pi \nu x}$

where the complex exponentials have amplitudes of 2 and frequencies of v and −v. See Figure 1.1. (To see how Figure 1.1(b) was produced, consult Exercise 1.3.)

Or, suppose our wave form is described by 6 sin 2π vx, which also has frequency v. Using Euler’s identity (1.1) again, we obtain

$6sin2\pi \nu x=3i{e}^{-i2\pi \nu x}-3i{e}^{i2\pi \nu x}$

where the complex exponentials have complex amplitudes of 3i and −3i and frequencies of −v and v. See Figure 1.2. (To see how Figure 1.2(b) was produced, consult Exercise 1.4.)

FIGURE 1.1
Frequency representation of 4 cos(2π vx), v = 9. (a) Graph in x domain (ti...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1 Basic Aspects of Fourier Series
• 2 The Discrete Fourier Transform (DFT)
• 3 The Fast Fourier Transform (FFT)
• 4 Some Applications of Fourier Series
• 5 Fourier Transforms
• 6 Fourier Optics
• A User’s Manual for Fourier Analysis Software
• B Some Computer Programs
• C The Schwarz Inequality
• D Solutions to Odd-Numbered Exercises
• Bibliography
• Index