Fast Fourier Transforms, Second Edition

This text provides a clear treatment of Fourier series, Fourier tranforms, and FFTs. The sofware included with the book and updated for this edition enables you to generate images of Fourier analysis described in the text. Topics include applications to vibrating strings (including computer animation), heat conduction, removal of noise and frequency detection, filtering of Fourier series and improvement of convergence, electron diffraction, imaging with coherent and incoherent monochromatic light, spectral analysis with diffraction gartings, Fourier transform properties of lenses, Poisson summation, sampling theory and aliasing.

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 SeriesDefinition of Fourier SeriesExamples of Fourier SeriesFourier Series of Real FunctionsPointwise Convergence of Fourier SeriesFurther Aspects of Convergence of Fourier SeriesFourier Sine Series and Cosine SeriesConvergence of Fourier Sine and Cosine SeriesReferencesExercisesChapter 2. The Discrete Fourier Transform (DFT)Derivation of the DFTBasic Properties of the DFTRelation of the DFT to Fourier CoefficientsRelation of the DFT to Sampled Fourier SeriesDiscrete Sine and Cosine TransformsReferencesExercisesChapter 3. The Fast Fourier Transform (FFT)Decimation in Time, Radix 2, FFTBit ReversalRotations in FFTsComputation of Sines and TangentsComputing Two Real FFTs SimultaneouslyComputing a Real FFTFast Sine and Cosine TransformsInversion of Discrete Sine and Cosine TransformsInversion of the FFT of a Real SequenceReferencesExercisesChapter 4. Some Applications of Fourier SeriesHeat EquationThe Wave EquationSchrödinger's Equation for a Free ParticleFilters Used in Signal ProcessingDesigning FiltersConvolution and Point Spread FunctionsDiscrete Convolutions Using FFTsKernels for Some Common FiltersConvergence of Filtered Fourier SeriesFurther Analysis of Fourier Series Partial SumsReferencesExercisesChapter 5. Fourier TransformsIntroductionProperties of Fourier TransformsInversion of Fourier TransformsThe Relation between Fourier Transforms and DFTsConvolution-An IntroductionThe Convolution TheoremAn Application of Convolution in Quantum MechanicsFiltering, Frequency Detection, and Removal of NoisePoisson SummationSummation Kernels Arising from Poisson SummationThe Sampling TheoremAliasingComparison of Three KernelsSine and Cosine TransformsReferencesExercisesChapter 6. Fourier OpticsIntroduction - Diffraction and Coherency of LightFresnel DiffractionFraunhofer DiffractionCircular AperturesInterferenceDiffraction GratingsSpectral Analysis with Diffraction GratingsThe Phase Transformation Induced by a Thin LensImaging with a Single LensImaging with Coherent LightFourier Transforming Property of a LensImaging with Incoherent LightReferencesExercisesA. User's Manual for Fourier Analysis SoftwareB. Some Computer ProgramsC. The Schwarz InequalityD. Solutions to Odd-Numbered ExercisesBibliographyIndex

Walker, James S.