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

423 Pages·2008·3.04 MB·English

by  Eleanor Chu

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
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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