Discrete and Continuous Fourier Transforms_ Ana..

423 Pages·2008·3.04 MB·English

D C
ISCRETE AND ONTINUOUS
F T
OURIER RANSFORMS
ANALYSIS, APPLICATIONS
AND FAST ALGORITHMS

D C
ISCRETE AND ONTINUOUS
F T
OURIER RANSFORMS
ANALYSIS, APPLICATIONS
AND FAST ALGORITHMS
Eleanor Chu
University of Guelph
Guelph, Ontario, Canada
MATLAB® 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 MATLAB® software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular
use of the MATLAB® 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
Contents
ListofFigures xi
ListofTables xv
Preface xvii
Acknowledgments xxi
AbouttheAuthor xxiii
I Fundamentals, AnalysisandApplications 1
1 AnalyticalandGraphicalRepresentationofFunctionContents 3
1.1 TimeandFrequencyContentsofaFunction . . . . . . . . . . . . . . . . . . 3
1.2 TheFrequency-DomainPlotsasGraphicalTools . . . . . . . . . . . . . . . 4
1.3 IdentifyingtheCosineandSineModes . . . . . . . . . . . . . . . . . . . . . 6
1.4 UsingComplexExponentialModes . . . . . . . . . . . . . . . . . . . . . . 7
1.5 UsingCosineModeswithPhaseorTimeShifts . . . . . . . . . . . . . . . . 9
1.6 PeriodicityandCommensurateFrequencies . . . . . . . . . . . . . . . . . . 12
1.7 ReviewofResultsandTechniques . . . . . . . . . . . . . . . . . . . . . . . 13
1.7.1 Practicingthetechniques . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 ExpressingSingleComponentSignals . . . . . . . . . . . . . . . . . . . . . 19
1.9 GeneralFormofaSinusoidinSignalApplication . . . . . . . . . . . . . . . 20
1.9.1 Expressingsequencesofdiscrete-timesamples . . . . . . . . . . . . 21
1.9.2 Periodicityofsinusoidalsequences . . . . . . . . . . . . . . . . . . 22
1.10 FourierSeries:ATopictoCome . . . . . . . . . . . . . . . . . . . . . . . . 23
1.11 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 SamplingandReconstructionofFunctions–PartI 27
2.1 DFTandBand-LimitedPeriodicSignal . . . . . . . . . . . . . . . . . . . . 27
2.2 FrequenciesAliasedbySampling. . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Connection:Anti-AliasingFilter . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 AlternateNotationsandFormulas . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 SamplingPeriodandAlternateFormsofDFT . . . . . . . . . . . . . . . . . 38
2.6 SampleSizeandAlternateFormsofDFT . . . . . . . . . . . . . . . . . . . 41
v
vi CONTENTS
3 TheFourierSeries 45
3.1 FormalExpansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Time-LimitedFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 EvenandOddFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Half-RangeExpansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 FourierSeriesUsingComplexExponentialModes. . . . . . . . . . . . . . . 60
3.6 Complex-ValuedFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 FourierSeriesinOtherVariables . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 TruncatedFourierSeriesandLeastSquares . . . . . . . . . . . . . . . . . . 61
3.9 OrthogonalProjectionsandFourierSeries . . . . . . . . . . . . . . . . . . . 63
3.9.1 TheCauchy(cid:150)Schw arzinequality . . . . . . . . . . . . . . . . . . . . 68
3.9.2 TheMinkowskiinequality . . . . . . . . . . . . . . . . . . . . . . . 71
3.9.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9.4 Least-squaresapproximation . . . . . . . . . . . . . . . . . . . . . . 74
3.9.5 Bessel(cid:146)sinequalityandRiemann(cid:146)slemma . . . . . . . . . . . . . . . 77
3.10 ConvergenceoftheFourierSeries . . . . . . . . . . . . . . . . . . . . . . . 79
3.10.1 Startingwithaconcreteexample . . . . . . . . . . . . . . . . . . . . 79
3.10.2 Pointwiseconvergence(cid:151) alocalproperty . . . . . . . . . . . . . . . 82
3.10.3 Therateofconvergence(cid:151) aglobalproperty . . . . . . . . . . . . . . 87
3.10.4 TheGibbsphenomenon . . . . . . . . . . . . . . . . . . . . . . . . 89
3.10.5 TheDirichletkernelperspective . . . . . . . . . . . . . . . . . . . . 91
3.10.6 EliminatingtheGibbseffectbytheCesarosum . . . . . . . . . . . . 95
3.10.7 ReducingtheGibbseffectbyLanczossmoothing . . . . . . . . . . . 99
3.10.8 Themodi(cid:30) cationofFourierseriescoef(cid:30) cients. . . . . . . . . . . . . 100
3.11 AccountingforAliasedFrequenciesinDFT . . . . . . . . . . . . . . . . . . 102
3.11.1 Samplingfunctionswithjumpdiscontinuities . . . . . . . . . . . . . 104
4 DFTandSampledSignals 109
4.1 DerivingtheDFTandIDFTFormulas . . . . . . . . . . . . . . . . . . . . . 109
4.2 DirectConversionBetweenAlternateForms . . . . . . . . . . . . . . . . . . 114
4.3 DFTofConcatenatedSampleSequences . . . . . . . . . . . . . . . . . . . . 116
4.4 DFTCoefc(cid:30) ientsofaCommensurateSum . . . . . . . . . . . . . . . . . . . 117
4.4.1 DFTcoef(cid:30) cientsofsingle-componentsignals . . . . . . . . . . . . . 117
4.4.2 Makingdirectuseofthedigitalfrequencies . . . . . . . . . . . . . . 121
4.4.3 Commonperiodofsampledcompositesignals . . . . . . . . . . . . 123
4.5 FrequencyDistortionbyLeakage . . . . . . . . . . . . . . . . . . . . . . . . 126
4.5.1 Fourierseriesexpansionofanonharmoniccomponent . . . . . . . . 128
4.5.2 AliasedDFTcoef(cid:30) cientsofanonharmoniccomponent . . . . . . . . 129
4.5.3 Demonstratingleakagebynumericalexperiments . . . . . . . . . . . 131
4.5.4 Mismatchingperiodicextensions. . . . . . . . . . . . . . . . . . . . 131
4.5.5 Minimizingleakageinpractice . . . . . . . . . . . . . . . . . . . . . 134
4.6 TheEffectsofZeroPadding . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.6.1 Zeropaddingthesignal . . . . . . . . . . . . . . . . . . . . . . . . . 134
CONTENTS vii
4.6.2 ZeropaddingtheDFT . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.7 ComputingDFTDen(cid:30) ingFormulasPerSe. . . . . . . . . . . . . . . . . . . 147
4.7.1 ProgrammingDFTinMATLAB(cid:1)R . . . . . . . . . . . . . . . . . . . 147
5 SamplingandReconstructionofFunctions–PartII 157
5.1 SamplingNonperiodicBand-LimitedFunctions . . . . . . . . . . . . . . . . 158
5.1.1 Fourierseriesoffrequency-limitedX(f) . . . . . . . . . . . . . . . 159
5.1.2 InverseFouriertransformoffrequency-limitedX(f) . . . . . . . . . 159
5.1.3 Recoveringthesignalanalytically . . . . . . . . . . . . . . . . . . . 160
5.1.4 Furtherdiscussionofthesamplingtheorem . . . . . . . . . . . . . . 161
5.2 DerivingtheFourierTransformPair . . . . . . . . . . . . . . . . . . . . . . 162
5.3 TheSineandCosineFrequencyContents . . . . . . . . . . . . . . . . . . . 164
5.4 TabulatingTwoSetsofFundamentalFormulas. . . . . . . . . . . . . . . . . 165
5.5 ConnectionswithTime/FrequencyRestrictions . . . . . . . . . . . . . . . . 165
5.5.1 ExamplesofFouriertransformpair . . . . . . . . . . . . . . . . . . 167
5.6 FourierTransformProperties . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.6.1 Derivingtheproperties . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.6.2 Utilitiesoftheproperties . . . . . . . . . . . . . . . . . . . . . . . . 175
5.7 AlternateFormoftheFourierTransform . . . . . . . . . . . . . . . . . . . . 177
5.8 ComputingtheFourierTransformfromDiscrete-TimeSamples . . . . . . . . 178
5.8.1 Almosttime-limitedandband-limitedfunctions . . . . . . . . . . . . 179
5.9 ComputingtheFourierCoef(cid:30) cientsfromDiscrete-TimeSamples . . . . . . . 181
5.9.1 Periodicandalmostband-limitedfunction . . . . . . . . . . . . . . . 182
6 SamplingandReconstructionofFunctions–PartIII 185
6.1 ImpulseFunctionsandTheirProperties . . . . . . . . . . . . . . . . . . . . 185
6.2 GeneratingtheFourierTransformPairs . . . . . . . . . . . . . . . . . . . . 188
6.3 ConvolutionandFourierTransform . . . . . . . . . . . . . . . . . . . . . . 189
6.4 PeriodicConvolutionandFourierSeries . . . . . . . . . . . . . . . . . . . . 192
6.5 ConvolutionwiththeImpulseFunction. . . . . . . . . . . . . . . . . . . . . 194
6.6 ImpulseTrainasaGeneralizedFunction . . . . . . . . . . . . . . . . . . . . 195
6.7 ImpulseSamplingofContinuous-TimeSignals . . . . . . . . . . . . . . . . 202
6.8 NyquistSamplingRateRediscovered. . . . . . . . . . . . . . . . . . . . . . 203
6.9 SamplingTheoremforBand-LimitedSignal . . . . . . . . . . . . . . . . . . 207
6.10 SamplingofBand-PassSignals . . . . . . . . . . . . . . . . . . . . . . . . . 209
7 FourierTransformofaSequence 211
7.1 DerivingtheFourierTransformofaSequence . . . . . . . . . . . . . . . . . 211
7.2 PropertiesoftheFourierTransformofaSequence . . . . . . . . . . . . . . . 215
7.3 GeneratingtheFourierTransformPairs . . . . . . . . . . . . . . . . . . . . 217
7.3.1 TheKroneckerdeltasequence . . . . . . . . . . . . . . . . . . . . . 217
7.3.2 RepresentingsignalsbyKroneckerdelta . . . . . . . . . . . . . . . . 218
7.3.3 Fouriertransformpairs . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.4 DualityinConnectionwiththeFourierSeries . . . . . . . . . . . . . . . . . 226
viii CONTENTS
7.4.1 Periodicconvolutionanddiscreteconvolution . . . . . . . . . . . . . 227
7.5 TheFourierTransformofaPeriodicSequence . . . . . . . . . . . . . . . . . 229
7.6 TheDFTInterpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.6.1 TheinterpretedDFTandtheFouriertransform . . . . . . . . . . . . 234
7.6.2 Time-limitedcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.6.3 Band-limitedcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.6.4 Periodicandband-limitedcase . . . . . . . . . . . . . . . . . . . . . 237
8 TheDiscreteFourierTransformofaWindowedSequence 239
8.1 ARectangularWindowofIn(cid:30) niteWidth . . . . . . . . . . . . . . . . . . . . 239
8.2 ARectangularWindowofAppropriateFiniteWidth . . . . . . . . . . . . . . 241
8.3 FrequencyDistortionbyImproperTruncation . . . . . . . . . . . . . . . . . 243
8.4 WindowingaGeneralNonperiodicSequence . . . . . . . . . . . . . . . . . 244
8.5 Frequency-DomainPropertiesofWindows . . . . . . . . . . . . . . . . . . . 245
8.5.1 Therectangularwindow . . . . . . . . . . . . . . . . . . . . . . . . 246
8.5.2 Thetriangularwindow . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.5.3 ThevonHannwindow . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.5.4 TheHammingwindow . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.5.5 TheBlackmanwindow . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.6 ApplicationsoftheWindowedDFT . . . . . . . . . . . . . . . . . . . . . . 252
8.6.1 Severalscenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8.6.2 SelectingthelengthofDFTinpractice . . . . . . . . . . . . . . . . 263
9 DiscreteConvolutionandtheDFT 267
9.1 LinearDiscreteConvolution . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.1.1 Linearconvolutionoftwon(cid:30) itesequences. . . . . . . . . . . . . . . 267
9.1.2 Sectioningalongsequenceforlinearconvolution . . . . . . . . . . . 273
9.2 PeriodicDiscreteConvolution . . . . . . . . . . . . . . . . . . . . . . . . . 273
9.2.1 De(cid:30)n itionbasedontwoperiodicsequences . . . . . . . . . . . . . . 273
9.2.2 Convertinglineartoperiodicconvolution . . . . . . . . . . . . . . . 275
9.2.3 De(cid:30) ningtheequivalentcyclicconvolution . . . . . . . . . . . . . . . 275
9.2.4 Thecyclicconvolutioninmatrixform . . . . . . . . . . . . . . . . . 278
9.2.5 Convertinglineartocyclicconvolution . . . . . . . . . . . . . . . . 280
9.2.6 Twocyclicconvolutiontheorems. . . . . . . . . . . . . . . . . . . . 280
9.2.7 Implementingsectionedlinearconvolution . . . . . . . . . . . . . . 283
9.3 TheChirpFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.3.1 Thescenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.3.2 Theequivalentpartiallinearconvolution. . . . . . . . . . . . . . . . 285
9.3.3 Theequivalentpartialcyclicconvolution . . . . . . . . . . . . . . . 286
10 ApplicationsoftheDFTinDigitalFilteringandFilters 291
10.1 TheBackground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
10.2 Application-OrientedTerminology . . . . . . . . . . . . . . . . . . . . . . . 292
10.3 RevisitGibbsPhenomenonfromtheFilteringViewpoint . . . . . . . . . . . 294
CONTENTS ix
10.4 ExperimentingwithDigitalFilteringandFilterDesign . . . . . . . . . . . . 296
II FastAlgorithms 303
11 IndexMappingandMixed-RadixFFTs 305
11.1 AlgebraicDFTversusFFT-ComputedDFT . . . . . . . . . . . . . . . . . . 305
11.2 TheRoleofIndexMapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.2.1 Thedecouplingprocess(cid:151) StageI . . . . . . . . . . . . . . . . . . . 307
11.2.2 Thedecouplingprocess(cid:151) StageII . . . . . . . . . . . . . . . . . . . 309
11.2.3 Thedecouplingprocess(cid:151) StageIII . . . . . . . . . . . . . . . . . . . 311
11.3 TheRecursiveEquationApproach . . . . . . . . . . . . . . . . . . . . . . . 313
11.3.1 CountingshortDFTorDFT-liketransforms . . . . . . . . . . . . . . 313
11.3.2 TherecursiveequationforarbitrarycompositeN . . . . . . . . . . . 313
11.3.3 Specializationtotheradix-2DITFFTforN =2ν . . . . . . . . . . 315
11.4 OtherFormsbyAlternateIndexSplitting. . . . . . . . . . . . . . . . . . . . 317
11.4.1 TherecursiveequationforarbitrarycompositeN . . . . . . . . . . . 318
11.4.2 Specializationtotheradix-2DIFFFTforN =2ν . . . . . . . . . . . 319
12 KroneckerProductFactorizationandFFTs 321
12.1 ReformulatingtheTwo-FactorMixed-RadixFFT . . . . . . . . . . . . . . . 322
12.2 FromTwo-FactortoMulti-FactorMixed-RadixFFT . . . . . . . . . . . . . . 328
12.2.1 SelectedpropertiesandrulesforKroneckerproducts . . . . . . . . . 329
12.2.2 CompletefactorizationoftheDFTmatrix . . . . . . . . . . . . . . . 331
12.3 OtherFormsbyAlternateIndexSplitting. . . . . . . . . . . . . . . . . . . . 333
12.4 FactorizationResultsbyAlternateExpansion . . . . . . . . . . . . . . . . . 335
12.4.1 Unorderedmixed-radixDITFFT. . . . . . . . . . . . . . . . . . . . 335
12.4.2 Unorderedmixed-radixDIFFFT . . . . . . . . . . . . . . . . . . . . 337
12.5 UnorderedFFTforScrambledInput . . . . . . . . . . . . . . . . . . . . . . 337
12.6 UtilitiesoftheKroneckerProductFactorization . . . . . . . . . . . . . . . . 339
13 TheFamilyofPrimeFactorFFTAlgorithms 341
13.1 ConnectingtheRelevantIdeas . . . . . . . . . . . . . . . . . . . . . . . . . 342
13.2 DerivingtheTwo-FactorPFA . . . . . . . . . . . . . . . . . . . . . . . . . . 343
13.2.1 StageI:Nonstandardindexmappingschemes . . . . . . . . . . . . . 343
13.2.2 StageII:DecouplingtheDFTcomputation . . . . . . . . . . . . . . 345
13.2.3 OrganizingthePFAcomputation(cid:150)P art1 . . . . . . . . . . . . . . . . 346
13.3 MatrixFormulationoftheTwo-FactorPFA . . . . . . . . . . . . . . . . . . 348
13.3.1 StageIII:TheKroneckerproductfactorization . . . . . . . . . . . . 348
13.3.2 StageIV:De(cid:30) ningpermutationmatrices . . . . . . . . . . . . . . . . 348
13.3.3 StageV:Completingthematrixfactorization . . . . . . . . . . . . . 350
13.4 MatrixFormulationoftheMulti-FactorPFA . . . . . . . . . . . . . . . . . . 350
13.4.1 OrganizingthePFAcomputation(cid:151) Part2 . . . . . . . . . . . . . . . 352
13.5 NumberTheoryandIndexMappingbyPermutations . . . . . . . . . . . . . 353