2nd Edition

# Fast Fourier Transforms

By James S. Walker Copyright 1996
464 Pages
by CRC Press

464 Pages
by CRC Press

Also available as eBook on:

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
SchrÃ¶dinger's Equation for a Free Particle
Filters Used in Signal Processing
Designing Filters
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

### Biography

Walker, James S.