2nd Edition

Fast Fourier Transforms

By James S. Walker Copyright 1996

    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
    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
    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
    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
    Chapter 5. Fourier Transforms
    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
    Comparison of Three Kernels
    Sine and Cosine Transforms
    Chapter 6. Fourier Optics
    Introduction - Diffraction and Coherency of Light
    Fresnel Diffraction
    Fraunhofer Diffraction
    Circular Apertures
    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
    A. User's Manual for Fourier Analysis Software
    B. Some Computer Programs
    C. The Schwarz Inequality
    D. Solutions to Odd-Numbered Exercises


    Walker, James S.