Download Discrete and continuous Fourier transforms: analysis, applications and fast algorithms PDF

Discrete and continuous Fourier transforms: analysis, applications and fast algorithms
Name: Discrete and continuous Fourier transforms: analysis, applications and fast algorithms
Author: eleanor chu
Pages: 423
Year: 2008
Language: English
File Size: 3.04 MB
Downloads: 0
Page 5

MATLABfi is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLABfi software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLABfi software. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487 2742 " 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number 13: 978 1 4200 6363 9 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid ity of all materials or the consequences of their use. The Authors and Publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978 750 8400. CCC is a not for profit organization that provides licenses and registration for a variety of users. For orga nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com


Page 6

Contents List of Figuresxi List of Tablesxv Prefacexvii Acknowledgmentsxxi About the Au thorxxiii IFundamentals, Analysis and Applications1 1Analyticaland Graphical Representation of Function Contents 3 1.1TimeandFrequencyContentsofaFunction.................. 3 1.2TheFrequency DomainPlotsasGraphicalTools............... 4 1.3IdentifyingtheCosineandSine Modes..................... 6 1.4 Using Complex Exponential Modes...................... 7 1.5UsingCosine ModeswithPhaseorTimeShifts................ 9 1.6PeriodicityandCommensurateFrequencies.................. 12 1.7ReviewofResultsandTechniques....................... 13 1.7.1Practicingthe techniques........................ 15 1.8 Expressing Single Component Signals . . . . . ................ 19 1.9GeneralFormofaSinusoidinSignalApplication............... 20 1.9.1Expressingsequencesof discrete timesamples ............ 21 1.9.2Periodicityofsinusoidalsequences.................. 22 1.10 FourierSeries:ATopictoCome........................ 23 1.11Terminology................................... 25 2Sampling and Reconstruction of Functions Part I27 2.1DFTandBand LimitedPeriodicSignal.................... 27 2.2FrequenciesAliasedbySampling. ....................... 32 2.3 Connection: Anti Aliasing Filter . ....................... 36 2.4AlternateNotationsandFormulas....................... 36 2.5SamplingPeriodand AlternateFormsofDFT................. 38 2.6SampleSizeand AlternateFormsofDFT................... 41 v


Page 7

viCONTENTS 3TheFourier Series45 3.1FormalExpansions ............................... 45 3.1.1Examples................................ 48 3.2Time LimitedFunctions............................ 50 3.3EvenandOddFunctions............................ 51 3.4Half Range Expansions............................. 53 3.5 Fourier Series Using Complex Exponential Modes............... 60 3.6Complex ValuedFunctions ........................... 60 3.7FourierSeriesinOtherVariables........................ 61 3.8Truncated FourierSeriesandLeastSquares.................. 61 3.9 Orthogonal Projections and Fourier Series . . . ................ 63 3.9.1 The Cauchy Schw arz inequality . . . . ................ 68 3.9.2TheMinkowski inequality ....................... 71 3.9.3Projections ............................... 72 3.9.4Least squaresapproximation...................... 74 3.9.5Bessel s inequality and Riemann s lemma............... 77 3.10Convergenceofthe FourierSeries....................... 79 3.10.1Startingwithaconcreteexample.................... 79 3.10.2Pointwiseconvergence alocalproperty............... 82 3.10.3Therateofconvergence aglobalproperty.............. 87 3.10.4TheGibbs phenomenon........................ 89 3.10.5TheDirichletkernelperspective.................... 91 3.10.6EliminatingtheGibbseffectbytheCesarosum ............ 95 3.10.7ReducingtheGibbseffectbyLanczossmoothing........... 99 3.10.8The modi36cationofFourierseriescoef36cients.............100 3.11Accountingfor AliasedFrequenciesinDFT..................102 3.11.1Sampling functions with jump discontinuities . ............104 4DFTandSampled Signals109 4.1DerivingtheDFTandIDFTFormulas.....................109 4.2DirectConversionBetween AlternateForms..................114 4.3DFTof ConcatenatedSampleSequences....................116 4.4DFTCoef36cientsofaCommensurateSum ...................117 4.4.1 DFT coef36 cients of single componentsignals . ............117 4.4.2Makingdirectuseofthe digitalfrequencies..............121 4.4.3Commonperiodofsampled compositesignals ............123 4.5FrequencyDistortionbyLeakage........................126 4.5.1 Fourier series expansion of a nonharmonic component ........128 4.5.2 Aliased DFT coef36 cients of a nonharmonic component . . . . . . . . 129 4.5.3Demonstratingleakage by numerical experiments...........131 4.5.4Mismatching periodicextensions....................131 4.5.5Minimizingleakageinpractice.....................134 4.6TheEffectsofZeroPadding..........................134 4.6.1Zeropaddingthesignal.........................134


Page 8

CONTENTSvii 4.6.2ZeropaddingtheDFT.........................141 4.7ComputingDFTDe36ningFormulasPerSe ...................147 4.7.1 Programming DFT in MATLAB R1 ...................147 5Sampling and Reconstruction of Functions Part II157 5.1 Sampling Nonperiodic Band Limited Functions ................158 5.1.1 Fourier series of frequency limitedX(f)...............159 5.1.2 Inverse Fourier transform of frequency limitedX(f).........159 5.1.3Recoveringthesignalanalytically ...................160 5.1.4Furtherdiscussion ofthesamplingtheorem..............161 5.2Derivingthe FourierTransformPair......................162 5.3TheSineandCosineFrequencyContents ...................164 5.4 Tabulating Two Sets of Fundamental Formulas . ................165 5.5Connections withTime/FrequencyRestrictions ................165 5.5.1ExamplesofFouriertransformpair..................167 5.6FourierTransformProperties..........................171 5.6.1Derivingthe properties.........................172 5.6.2Utilities of the properties . .......................175 5.7AlternateFormofthe FourierTransform....................177 5.8Computingthe FourierTransform fromDiscrete TimeSamples........178 5.8.1Almosttime limitedand band limitedfunctions ............179 5.9Computingthe FourierCoef36cientsfromDiscrete TimeSamples.......181 5.9.1Periodicandalmostband limitedfunction...............182 6Sampling and Reconstruction of Functions Part III185 6.1Impulse FunctionsandTheirProperties....................185 6.2Generatingthe FourierTransformPairs....................188 6.3Convolutionand FourierTransform......................189 6.4PeriodicConvolutionand FourierSeries....................192 6.5Convolutionwith theImpulse Function.....................194 6.6ImpulseTrainasaGeneralized Function....................195 6.7 Impulse Sampling of Continuous Time Signals ................202 6.8 Nyquist Sampling Rate Rediscovered......................203 6.9SamplingTheoremforBand LimitedSignal..................207 6.10SamplingofBand PassSignals.........................209 7Fourier Transform of a Sequence211 7.1Derivingthe FourierTransformofaSequence.................211 7.2Propertiesofthe FourierTransformofaSequence...............215 7.3Generatingthe FourierTransformPairs....................217 7.3.1TheKroneckerdeltasequence.....................217 7.3.2RepresentingsignalsbyKroneckerdelta ................218 7.3.3Fouriertransformpairs.........................219 7.4 Duality in Connection with the Fourier Series . ................226


Page 9

viiiCONTENTS 7.4.1Periodicconvolutionand discreteconvolution.............227 7.5The FourierTransformofaPeriodicSequence.................229 7.6TheDFTInterpretation.............................232 7.6.1TheinterpretedDFTandthe Fouriertransform ............234 7.6.2Time limitedcase............................235 7.6.3Band limitedcase............................236 7.6.4Periodicand band limitedcase.....................237 8TheDiscreteFourier Transform of a Windowed Sequence239 8.1 A Rectangular Window of In36 nite Width . . . . ................239 8.2 A Rectangular Window of Appropriate Finite Width . . . . . . ........241 8.3FrequencyDistortionbyImproperTruncation.................243 8.4 Windowing a General Nonperiodic Sequence . ................244 8.5Frequency DomainPropertiesofWindows ...................245 8.5.1 The rectangular window . .......................246 8.5.2 The triangular window . . .......................247 8.5.3ThevonHannwindow.........................248 8.5.4TheHammingwindow.........................250 8.5.5TheBlackmanwindow.........................251 8.6Applications oftheWindowedDFT......................252 8.6.1Several scenarios............................252 8.6.2SelectingthelengthofDFTinpractice................263 9DiscreteConvolutionand the DFT 267 9.1LinearDiscreteConvolution..........................267 9.1.1Linearconvolutionoftwo36nitesequences...............267 9.1.2Sectioningalongsequenceforlinearconvolution...........273 9.2PeriodicDiscreteConvolution.........................273 9.2.1De36n ition based on two periodic sequences . . ............273 9.2.2Converting lineartoperiodicconvolution...............275 9.2.3De36ningtheequivalentcyclicconvolution...............275 9.2.4Thecyclicconvolutioninmatrixform.................278 9.2.5Converting lineartocyclicconvolution................280 9.2.6Twocyclicconvolutiontheorems. ...................280 9.2.7Implementingsectionedlinear convolution..............283 9.3TheChirpFourierTransform..........................284 9.3.1Thescenario..............................284 9.3.2The equivalent partial linear convolution................285 9.3.3The equivalent partialcyclicconvolution...............286 10 Applications of theDFTinDigitalFiltering and Filters 291 10.1 The Background . . ...............................291 10.2Application OrientedTerminology .......................292 10.3Revisit Gibbs Phenomenon from the Filtering Viewpoint...........294


Page 10

CONTENTSix 10.4Experimenting with Digital Filtering and Filter Design ............296 II Fast Algorithms303 11 Index Mapping and Mixed Radix FFTs305 11.1AlgebraicDFTversusFFT ComputedDFT..................305 11.2TheRoleof IndexMapping ...........................306 11.2.1The decouplingprocess StageI ...................307 11.2.2The decouplingprocess StageII...................309 11.2.3The decouplingprocess StageIII ...................311 11.3The RecursiveEquationApproach .......................313 11.3.1CountingshortDFTorDFT liketransforms..............313 11.3.2 The recursive equation for arbitrary compositeN...........313 11.3.3 Specialization to the radix 2 DIT FFT forN=2 1 ..........315 11.4Other Forms by Alternate Index Splitting . . . . ................317 11.4.1 The recursive equation for arbitrary compositeN...........318 11.4.2 Specialization to the radix 2 DIF FFT forN=2 1 ...........319 12 Kronecker Product Factorization and FFTs321 12.1ReformulatingtheTwo FactorMixed RadixFFT...............322 12.2FromTwo Factor to Multi Factor Mixed Radix FFT . . . . . . ........328 12.2.1 Selected properties and rules for Kronecker products . ........329 12.2.2CompletefactorizationoftheDFTmatrix...............331 12.3Other Forms by Alternate Index Splitting . . . . ................333 12.4FactorizationResultsby AlternateExpansion.................335 12.4.1Unorderedmixed radixDITFFT....................335 12.4.2Unorderedmixed radixDIFFFT....................337 12.5 Unordered FFT for Scrambled Input......................337 12.6 Utilities of the Kr onecker Product Factorization . . . . ............339 13 The Family of Prime Factor FFT Algorithms341 13.1 Connecting the Relevant Ideas . . .......................342 13.2DerivingtheTwo FactorPFA..........................343 13.2.1StageI: Nonstandard indexmappingschemes.............343 13.2.2StageII:DecouplingtheDFTcomputation..............345 13.2.3 Organizing the PFA computation P art1 ................346 13.3MatrixFormulationoftheTwo FactorPFA..................348 13.3.1 Stage III: The Kronecker product factorization . . . . ........348 13.3.2StageIV:De36ning permutationmatrices................348 13.3.3StageV:Completingthematrixfactorization.............350 13.4Matrix Formulation of the Multi Factor PFA . . ................350 13.4.1OrganizingthePFAcomputation Part2...............352 13.5NumberTheoryand IndexMapping byPermutations.............353


Tags: Download Discrete and continuous Fourier transforms: analysis, applications and fast algorithms PDF, Discrete and continuous Fourier transforms: analysis, applications and fast algorithms free pdf download, Discrete and continuous Fourier transforms: analysis, applications and fast algorithms Pdf online download, Discrete and continuous Fourier transforms: analysis, applications and fast algorithms By eleanor chu download, Discrete and continuous Fourier transforms: analysis, applications and fast algorithms.pdf, Discrete and continuous Fourier transforms: analysis, applications and fast algorithms read online.
About | Contact | DMCA | Terms | Privacy | Mobile Specifications
Copyright 2021 FilePdf