  # Fast Fourier Transforms

## 2nd Edition

CRC Press

464 pages

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pub: 1996-08-22
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### Description

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 :

Chapter 1. Basic Aspects of Fourier Series

Definition of Fourier Series

Examples of Fourier Series

Fourier Series of Real Functions

Pointwise Convergence of Fourier Series

Further Aspects of Convergence of Fourier Series

Fourier Sine Series and Cosine Series

Convergence of Fourier Sine and Cosine Series

References

Exercises

Chapter 2. The Discrete Fourier Transform (DFT)

Derivation of the DFT

Basic Properties of the DFT

Relation of the DFT to Fourier Coefficients

Relation of the DFT to Sampled Fourier Series

Discrete Sine and Cosine Transforms

References

Exercises

Chapter 3. The Fast Fourier Transform (FFT)

Decimation in Time, Radix 2, FFT

Bit Reversal

Rotations in FFTs

Computation of Sines and Tangents

Computing Two Real FFTs Simultaneously

Computing a Real FFT

Fast Sine and Cosine Transforms

Inversion of Discrete Sine and Cosine Transforms

Inversion of the FFT of a Real Sequence

References

Exercises

Chapter 4. Some Applications of Fourier Series

Heat Equation

The Wave Equation

Schrödinger's Equation for a Free Particle

Filters Used in Signal Processing

Designing Filters

Convolution and Point Spread Functions

Discrete Convolutions Using FFTs

Kernels for Some Common Filters

Convergence of Filtered Fourier Series

Further Analysis of Fourier Series Partial Sums

References

Exercises

Chapter 5. Fourier Transforms

Introduction

Properties of Fourier Transforms

Inversion of Fourier Transforms

The Relation between Fourier Transforms and DFTs

Convolution-An Introduction

The Convolution Theorem

An Application of Convolution in Quantum Mechanics

Filtering, Frequency Detection, and Removal of Noise

Poisson Summation

Summation Kernels Arising from Poisson Summation

The Sampling Theorem

Aliasing

Comparison of Three Kernels

Sine and Cosine Transforms

References

Exercises

Chapter 6. Fourier Optics

Introduction - Diffraction and Coherency of Light

Fresnel Diffraction

Fraunhofer Diffraction

Circular Apertures

Interference

Diffraction Gratings

Spectral Analysis with Diffraction Gratings

The Phase Transformation Induced by a Thin Lens

Imaging with a Single Lens

Imaging with Coherent Light

Fourier Transforming Property of a Lens

Imaging with Incoherent Light

References

Exercises

A. User's Manual for Fourier Analysis Software

B. Some Computer Programs

C. The Schwarz Inequality

D. Solutions to Odd-Numbered Exercises

Bibliography

Index

### Subject Categories

##### BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT003000
MATHEMATICS / Applied