Digital signal processing : spectral computation and filter design

457 Pages·2001·6.871 MB·English

by  Chi-Tsong Chen

Digital Signal Processing
Spectral Computation and Filter Design
Chi-Tsong Chen
StateUniversityofNewYorkatStonyBrook
NewYork Oxford
OXFORDUNIVERSITYPRESS
2001
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LibraryofCongressCataloging-in-PublicationData
Chen,Chi-Tsong.
Digitalsignalprocessing:spectralcomputationandfilterdesign/Chi-TsongChen.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN0-19-513638-1
1.Signalprocessing—Digitaltechniques. 2.Electricfilters—Designandconstruction.
3.Spectralsequences(Mathematics) I.Title.
TK5102.9.C476 2000
621.382'2—dc21 00-027881
Printing(lastdigit):10987654321
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PREFACE
This text is intended for use in a first course on digital signal processing (DSP), typically in
seniororfirst-yeargraduatelevel.Itmayalsobeusefultoengineersandscientistswhousedigital
computers to process measured data. Some elementary knowledge on signals and systems is
helpful but is not necessary. An attempt has been made to make this text as self-contained as
possible.
Asanintroductorytext,wediscussonlytwomajortopics:computercomputationoffrequency
contentsofsignalsanddesignofdigitalfilters.Thesetwotopicsfollowlogicallyabasictext
or course on Signals and Systems, which now often covers both continuous-time (CT) and
discrete-time(DT)cases.Tobeself-contained,westartfromscratch.Chapter1introducesCT
signals,DTsignals,anddigitalsignals.WediscusswhyCTsignalsarenowwidelyprocessed
digitallyandhowtheyareconvertedintodigitalsignals.Wealsodiscusswhywestudyinthis
text, as in every other DSP text, only CT and DT signals even though digital signals are the
signalsprocessedondigitalcomputers.InChapter2,wefirstdiscussthefrequencyofCTand
DTsinusoidalsignals.WethenintroduceCTandDTFourierseries(frequencycomponents)for
periodicsignalsandestablishtheirrelationships.WethenusethefastFouriertransform(FFT)
tocomputetheirfrequencycomponents.
Chapter 3 extends CT and DT Fourier series to CT and DT Fourier transforms (frequency
spectra)thatcanrevealfrequencycontentsofaperiodicsignalsaswell.Weestablishthesampling
theoremanddiscusstheeffectsoftruncationofsignalsonfrequencyspectra.InChapter4,we
showthatcomputercomputationofDTFouriertransform(DTFT)leadsnaturallytothediscrete
Fouriertransform(DFT),whichdiffersfromtheDTFourierseriesonlybyafraction.Wethen
introduceaversionofFFT,anefficientwayofcomputingDFT.WeuseFFTtocomputefrequency
spectraofDTandCTsignalsandtocomputeDTandCTsignalsfromtheirfrequencyspectra.
Thiscompletesthediscussionofspectralcomputationofsignals.
Thesecondpartofthistextdiscussesthedesignofdigitalfilters.WeintroduceinChapter5
theclassofdigitalfilters(linear,time-invariant,lumped,causal,andstable)tobedesignedand
themathematicaltools(convolutions,impulseresponses, z-transform,andtransferfunctions)
tobeused.Wethenintroducetheconceptoffrequencyresponsesforstablesystems.Chapter
6discussesspecificationsofdigitalfiltersandhowtousepolesandzerostoshapemagnitude
responsesofsimpledigitalfilters.Chapter7introducesvariousmethodstodesignfinite-impulse-
response(FIR)digitalfilters.InChapter8,wefirstdiscussthereasonsfornotdesigningdirectly
infinite-impulse-response(IIR)digitalfilters.Wethenintroduceanalogprototypefilters,which
arelowpassfilterswith1rad/sastheirpassbandorstopbandedgefrequency.Allotheranalogand
alldigitalfrequency-selectivefilterscanthenbeobtainedbyusingfrequencytransformations.
xiii
xiv PREFACE
Thelastchapterdiscussesanumberofblockdiagramsfordigitalfiltersandsomeproblemsdue
tofinite-word-lengthimplementation.ThischapterisessentiallyindependentofChapters7and
8andmaybestudiedrightafterChapter6.
Althoughmosttopicsinthistextarestandard,ourpresentationandemphasisaresignificantly
differentfromotherexistingDSPtexts.Theyarediscussedinthefollowing.
• MostDSPtextsdiscusssystemsandsignalsintertwined.Thistextdiscussesinthefirstpart
onlysignalsandtheirspectralcomputation.Thusthediscussioncanbemorefocused.Itis
alsomoreefficientforthosewhowishtolearnhowtouseFFTtoanalyzemeasureddata.
• ThistextshowsthatthefrequencyofaDTsinusoidalsequencecannotbedefinedfromits
fundamentalperiodasintheCTcase.ItisthendefinedfromaCTsinusoidandjustified
usingaphysicalargument.Thusthistextusesthesamenotationtodenotefrequenciesin
CT and DT signals as opposed to different notations used in most DSP texts. We do use
different notations when the analog frequency range (−∞, ∞) is compressed into the
digitalfrequencyrange(−π, π]inbilineartransformations.
• AssumingthereadertohavehadFourieranalysisofCTsignals,mostDSPtextscoveronly
FourieranalysisofDTsignals.BecausesignalsprocessedbyDSParemostlyCT,thistext
coversFourieranalysesofbothCTandDTsignals.ThediscussionoftheCTpart,however,
isnotexhaustive.ItisintroducedtotheextenttoshowthedifferencesbetweenCTandDT
Fourier analyses and to establish their relationships. We establish sampling theorems for
puresinusoids,periodicsignals,andgeneralsignals.
• Most DSP texts assume the sampling period T to be 1 in DT Fourier analysis. Although
the equations involved are simpler, they become more complex in relating spectra of CT
signalsandtheirsampledsequences.ThistextdoesnotassumeT = 1anddiscusseshow
toselectT inspectralcomputationofCTsignalsfromtheirtimesamples.Indigitalfilter
design, however, we can select T to be 1 or any other value such as π in MATLAB,1 as
discussedinSection5.3.2.
• ManyDSPtextsintroduceDFTaftertheFouriertransformsanddiscussitasanindependent
mathematicalentity.OurdiscussionofDFTisbriefbecauseitisessentiallythesameasthe
DTFourierseries.Itisintroducedasacomputationaltool.
• Most DSP texts introduce first the two-sided z-transform and then reduce it to the one-
sided z-transform. This text does not need the two-sided transform. Thus we concentrate
on the one-sided z-transform and give reasons for forgoing the concept of the region of
convergence,whichisadifficultconceptandrarelyneededinapplication.
• Fourier analysis of DT systems are covered in most DSP texts. It is not covered in this
text because Fourier analysis of DT systems is less general and more complex than the
z-transform(Section5.5.3).
• Most DSP texts use exclusively negative-power transfer functions. This text uses both
negative-powerandpositive-powertransferfunctionsbecausethelatterismoreconvenient
in introducing the concepts of properness, degrees, poles, and zeros. Furthermore, both
formsareusedinMATLAB.
1MATLABisatrademarkoftheMathWorks,Inc.
PREFACE xv
• ThebilineartransformationintroducedinmanyDSPtextsnormalizethefrequencyrange
byassumingT =1andyetintroducethefactorT/2whereT maybedifferentfrom1.This
isinconsistent.Weintroduceanarbitraryfactorandshowthatthedesignisindependentof
thefactor.Thisismorelogicalandcanalsojustifytheinconsistence.
Inaddition,thistextdiscussesanumberoftopicsnotfoundinmostDSPtexts.Welistsomeof
theminthefollowing:
• Give a formal definition of the frequency of sinusoidal sequences and give a precise
frequencyrange(−π/T, π/T]forDTsignals(Section2.2).
• EstablishasimplifiedversionofthesamplingtheoremforperiodicsignalsandthenuseFFT
to compute frequency components of CT periodic signals from their sampled sequences.
DiscusshowtousetheMATLABfunctionsfftshift,ceil,floorandanewlydefinedfunction
shift to plot FFT computed frequency components in (−π/T, π/T] or [−π/T, π/T)
(Sections2.6and2.7).
• DiscussthenonuniquenessofinverseDFTandamethodofdeterminingthelocationofa
timesignalcomputedfromfrequencysamplesofitsfrequencyspectrum(Section4.7).
• UseFFTtocomputetheinversez-transform(Section5.4.2).
• Discusssteady-stateandtransientresponsesofdigitalfilters,andgiveanestimatedtimefor
atransientresponsetodieout(Section5.7.1).Comparetwowaysofintroducingtheconcept
offrequencyresponses.Wearguethatalthoughtheconceptcanbemoreeasilyintroduced
byassuminganinputtobeappliedfrom−∞,twoconceptsofpracticalimportancemay
oftenbeconcealed(Section5.7.3).
• Give a mathematical justification of using an antialiasing analog filter in digital signal
processing(Section6.7.1).
• IntroduceadiscreteleastsquaresmethodtodesignFIRfilters.Themethod,althoughvery
simple, does not seem to appear anywhere in the literature. The method is more flexible
thanthemethodoffrequencysamplingandleadsnaturallytotheMATLABfunctionfirls
(Sections7.6and7.6.1).
• Introduceananalogbandstoptransformationthatyieldsbetterbandstopfiltersthantheones
generatedbyMATLAB(Section8.4).
• Although FFT is very efficient in computing the convolution of two long sequences, we
givepossiblereasonsfornotusingFFTintheMATLABfunctionconv(Section9.4.1).
All terminology in this text is carefully defined. For example, both the Fourier series and
Fouriertransformsofperiodicsignalsareoftencalleddiscretefrequencyspectraintheliterature.
ThistextreservesfrequencyspectraexclusivelyfortheFouriertransforms,andcallsFourierse-
riesfrequencycomponents.Weattempttomakealldiscussionmathematicallycorrect;however,
wewillnotbeconstrainedbypuremathematics(Section3.9andthefootnoteinSection6.3).Asa
textintendedforpracticalapplication,thediscussionisnotnecessarilyrigorous.Forexample,we
usetheterms“veryclose”and“practicallyzero”loosely.Becauseoftheinformationexplosion,
weallhavelesstimetostudyasubjectarea.Thusthediscussioninthistextisnotexhaustive;we
concentrateonlyonconceptsandresultsthat,inouropinion,areessentialinpracticalapplication.
xvi PREFACE
MATLAB is an integral part of this text. We use while loops and if-else-end structures to
developMATLABprograms.However,ouremphasisisnotonMATLABbutratheronbasic
ideasandproceduresindigitalsignalprocessing.ThusthistextlistsonlyessentialMATLAB
functionsinmostprogramsandskipsfunctionsthatadjustshapesandsizesofplots,anddraw
horizontalandverticalcoordinates.Itisrecommendedthatthereaderrepeateachprogram.Even
thoughhemaynotobtainidenticalresults,allessentialinformationwillbegenerated.Clearly
anyotherDSPpackagecanalsobeused.
All numerical examples in the text and most problems at the end of each chapter are very
simple and can be solved analytically by hand. The reader is urged to do so. After obtaining
resultsbyhand,onecanthencomparethemwithcomputer-generatedresults.Thisisthebest
wayoflearningatopicandacomputationaltool.Oncemasteringthesubjectandtool,wecan
then apply the tool to real-world problems. A solutions manual, in which complete programs
arelistedforproblemsthatuseMATLAB,isavailablefromthepublisher.
This text is a complete restructure and rewriting of One-Dimensional Digital Signal Pro-
cessing,publishedin1979.Wedeletedthediscussionofthetwo-sidedz-transform,itsregion
ofconvergence,andthetopicssuchaserroranalyses,WienerFIR,andIIRfilters,thatrequire
statisticalmethods.Allaforementionedsections,exceptSection6.6.1,arenewinthistext.
I am indebted to many people in developing this text. First I like to thank Professors Petar
Djuric,AdrianLeuciuc,JohnMurray,NamPhamdo,StephenRappaport,KennethShort,and
ArmenZemanian,mycolleaguesatStonyBrook.IwenttothemwheneverIhadanyquestion
ordoubt.Manypeoplereviewedaearlierversionofthistext.Theircommentspromptedmeto
rewritemanysections.Ithankthemall.
IamgratefultomyeditorPeterC.Gordonforhisconfidenceinandenthusiasmaboutthis
project. Many people at Oxford University Press including Christine D’Antonio, Jacki Hartt,
AnitaSung,andJasmineUrmeneta,weremosthelpfulinthisundertaking.FinallyIthankmy
wife,Bih-Jau,forhersupport.
Chi-TsongChen
CONTENTS
Preface xiii
1 Introduction 1
1.1 Continuous-Time(CT),Discrete-Time(DT),andDigitalSignals 1
1.1.1 PlottingofDTSignals 3
1.2 RepresentationofDigitalSignals 4
1.3 A/DandD/AConversions 6
1.4 ComparisonofDigitalandAnalogTechniques 11
1.5 ApplicationsofDigitalSignalProcessing 12
1.6 ScopeoftheBook 14
1.6.1 SpectralComputation 15
1.6.2 DigitalFilterDesign 18
PART 1 Spectral Computation
2 CT and DT Fourier Series—Frequency Components 21
2.1 Introduction 21
2.1.1 FrequencyofCTSinusoids 22
2.2 FrequencyandFrequencyRangeofSinusoidalSequences 22
2.3 FrequenciesofCTSinusoidsandTheirSampledSequences 28
2.3.1 Applications 30
2.3.2 RecoveringaSinusoidfromItsSampledSequence 31
2.4 Continuous-TimeFourierSeries(CTFS) 34
2.4.1 DistributionofAveragePowerinFrequencies 42
2.4.2 ArePhasesImportant? 43
2.5 Discrete-TimeFourierSeries(DTFS) 45
2.5.1 RangeofFrequencyIndexm 51
2.5.2 TimeShifting 52
2.6 FFTComputationofDTFSCoefficients 54
2.6.1 RearrangingtheOutputoftheFFT 55
vii
viii CONTENTS
2.7 FFTComputationofCTFSCoefficients 59
2.7.1 FrequencyAliasingduetoTimeSampling 66
2.7.2 SelectingN toHaveNegligibleFrequencyAliasing 69
2.8 AveragePowerandItsComputation 75
2.9 ConcludingRemarks 78
Problems 78
3 CT and DT Fourier Transforms—Frequency Spectra 82
3.1 Introduction 82
3.2 CTFourierTransform(CTFT) 82
3.2.1 FrequencySpectrumofCTPeriodicSignals 89
3.3 PropertiesofFrequencySpectra 92
3.3.1 BoundednessandContinuity 92
3.3.2 EvenandOdd 93
3.3.3 TimeShifting 96
3.3.4 FrequencyShifting 96
3.3.5 TimeCompressionandExpansion 97
3.4 DistributionofEnergyinFrequencies 99
3.5 EffectsofTruncation 101
3.5.1 GibbsPhenomenon 103
3.6 DTFourierTransform(DTFT) 106
3.6.1 FrequencySpectrumofDTPeriodicSignals 113
3.7 EffectsofTruncation 114
3.8 NyquistSamplingTheorem 117
3.8.1 FrequencyAliasingduetoTimeSampling 122
3.9 Time-limitedBandlimitedTheorem 126
3.9.1 PracticalReconstructionofx(t)fromx(nT) 127
Problems 128
4 DFT and FFT—Spectral Computation 132
4.1 Introduction 132
4.2 DiscreteFourierTransform(DFT) 132
4.2.1 RelationshipbetweenDFTandDTFS 140
4.2.2 InverseDFTandInverseDTFT—TimeAliasingdueto
FrequencySampling 141
4.3 PropertiesofDFT 144
4.3.1 EvenandOdd 144
4.3.2 PeriodicShifting 145
4.4 FastFourierTransform(FFT) 146
4.4.1 OtherFFTandDSPProcessors 150
4.4.2 RealSequences 152
4.5 SpectralComputationofFiniteSequences 155
CONTENTS ix
4.5.1 PaddingwithZeros 156
4.5.2 SpectralComputationofInfiniteSequences 159
4.6 SpectralComputationofCTSignals 163
4.6.1 SpectralComputationofCTPeriodicSignals 171
4.7 ComputingDTSignalsfromSpectra 174
4.7.1 ComputingCTSignalsfromSpectra 179
4.8 ComputingEnergyUsingFFT 182
4.9 ConcludingRemarks 184
Problems 186
PART 2 Digital Filter Design
5 Linear Time-Invariant Lumped Systems 191
5.1 Introduction 191
5.2 LinearityandTimeInvariance 192
5.2.1 LTISystems—Convolutions 193
5.3 LTILSystems—DifferenceEquations 198
5.3.1 RecursiveandNonrecursiveDifferenceEquations 202
5.3.2 SamplingPeriodandReal-TimeProcessing 203
5.4 z-Transform 204
5.4.1 Inversez-Transform 210
5.4.2 FFTComputationoftheInversez-Transform 213
5.5 TransferFunctions 215
5.5.1 PolesandZeros 218
5.5.2 TransferFunctionsofFIRandIIRFilters 222
5.5.3 DTFourierTransformandz-Transform 223
5.6 Stability 225
5.6.1 TheJuryTest 226
5.7 FrequencyResponse 228
5.7.1 InfiniteTime 234
5.7.2 FrequencyResponseandFrequencySpectrum 235
5.7.3 AlternativeDerivationofFrequencyResponses 237
5.8 Continuous-TimeLTILSystems 239
5.8.1 LaplaceTransformandz-Transform 241
5.9 CTTransferFunction,Stability,andFrequencyResponse 242
5.9.1 MeasuringCTFrequencyResponses 245
5.10 ConcludingRemarks 247
Problems 248
6 Ideal and Some Practical Digital Filters 252
6.1 Introduction 252
x CONTENTS
6.2 IdealDigitalFilters 252
6.3 Realizability 253
6.3.1 FilterSpecifications 257
6.3.2 DigitalProcessingofAnalogSignals 259
6.4 First-OrderDigitalFilters 261
6.4.1 Second-OrderDigitalFilters 270
6.5 ReciprocalRootsandAll-PassFilters 275
6.6 MiscellaneousTopics 280
6.6.1 CombFilters 280
6.6.2 SinusoidGenerators 281
6.6.3 GoertzelAlgorithm 284
6.7 AnalogIdealLow-PassFilters 286
6.7.1 WhyAntialiasingFilters? 289
Problems 292
7 Design of FIR Filters 294
7.1 Introduction 294
7.2 ClassificationofLinear-PhaseFIRFilters 295
7.3 Least-SquaresOptimalFilters—DirectTruncation 302
7.4 WindowMethod 305
7.5 DesiredFilterswithSpecifiedTransitionBands 310
7.5.1 DesignbyFrequencySampling 313
7.6 DiscreteLeast-SquaresOptimalFIRFilters 316
7.6.1 IntegralLeast-SquaresOptimalFIRFilters 324
7.7 MinimaxOptimalFIRFilters 327
7.8 DesignofDigitalDifferentiators 332
7.9 HilbertTransformers 340
7.9.1 FromFIRLow-PassFilterstoHilbertTransformers 346
7.10 ADesignExample 349
Problems 352
8 Design of IIR Filters 355
8.1 Introduction 355
8.2 DifficultiesinDirectIIRFilterDesign 356
8.3 DesignofAnalogPrototypeFilters 358
8.4 AnalogFrequencyTransformations 364
8.5 ImpulseInvarianceMethod 374
8.5.1 DigitalFrequencyTransformations 379
8.6 BilinearTransformation 386
8.7 Analog-Prototype-to-DigitalTransformations 390
8.8 ComparisonswithFIRFilters 394
Problems 395