Table Of ContentDiscrete-Time Signal Processing
Alan V. Oppenheim Ronald W. Schafer
Third Edition
ISBN 10: 1-292-02572-7
ISBN 13: 978-1-292-02572-8
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ISBN 10: 1-292-02572-7
ISBN 13: 978-1-292-02572-8
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Introduction
Alan V. Oppenheim/Ronald W. Schafer 1
2. Discrete-Time Signals and Systems
Alan V. Oppenheim/Ronald W. Schafer 11
3. The z-Transform
Alan V. Oppenheim/Ronald W. Schafer 105
4. Sampling of Continuous-Time Signals
Alan V. Oppenheim/Ronald W. Schafer 163
5. Transform Analysis of Linear Time-Invariant Systems
Alan V. Oppenheim/Ronald W. Schafer 287
6. Structures for Discrete-Time Systems
Alan V. Oppenheim/Ronald W. Schafer 391
7. Filter Design Techniques
Alan V. Oppenheim/Ronald W. Schafer 517
8. The Discrete Fourier Transform
Alan V. Oppenheim/Ronald W. Schafer 651
9. Computation of the Discrete Fourier Transform
Alan V. Oppenheim/Ronald W. Schafer 747
10. Fourier Analysis of Signals Using the Discrete Fourier Transform
Alan V. Oppenheim/Ronald W. Schafer 827
11. Parametric Signal Modeling
Alan V. Oppenheim/Ronald W. Schafer 931
12. Discrete Hilbert Transforms
Alan V. Oppenheim/Ronald W. Schafer 985
Appendix: Random Signals
Alan V. Oppenheim/Ronald W. Schafer 1025
I
11003497
Appendix: Continuous-Time Filters
Alan V. Oppenheim/Ronald W. Schafer 1039
Index 1047
II
Introduction
Therichhistoryandfuturepromiseofsignalprocessingderivefromastrongsynergy
betweenincreasinglysophisticatedapplications,newtheoreticaldevelopmentsandcon-
stantlyemergingnewhardwarearchitecturesandplatforms.Signalprocessingapplica-
tions span an immense set of disciplines that include entertainment,communications,
spaceexploration,medicine,archaeology,geophysics,justtonameafew.Signalprocess-
ingalgorithmsandhardwareareprevalentinawiderangeofsystems,fromhighlyspe-
cializedmilitarysystemsandindustrialapplicationstolow-cost,high-volumeconsumer
electronics.Althoughweroutinelytakeforgrantedtheextraordinaryperformanceof
multimedia systems,such as high definition video,high fidelity audio,and interactive
games,these systems have always relied heavily on state-of-the-art signal processing.
Sophisticateddigitalsignalprocessorsareatthecoreofallmoderncellphones.MPEG
audioandvideoandJPEG1 imagedatacompressionstandardsrelyheavilyonsignal
processingprinciplesandtechniques.High-densitydatastoragedevicesandnewsolid-
statememoriesrelyincreasinglyontheuseofsignalprocessingtoprovideconsistency
and robustness to otherwise fragile technologies.As we look to the future,it is clear
that the role of signal processing is expanding,driven in part by the convergence of
communications,computers,andsignalprocessinginboththeconsumerarenaandin
advancedindustrialandgovernmentapplications.
The growing number of applications and demand for increasingly sophisticated
algorithmsgohand-in-handwiththerapiddevelopmentofdevicetechnologyforimple-
mentingsignalprocessingsystems.Bysomeestimates,evenwithimpendinglimitations
1TheacronymsMPEGandJPEGarethetermsusedinevencasualconversationforreferringtothe
standardsdevelopedbythe“MovingPictureExpertGroup(MPEG)”andthe“JointPhotographicExpert
Group(JPEG)”ofthe“InternationalOrganizationforStandardization(ISO).”
FromChapter1ofDiscrete-TimeSignalProcessing,ThirdEdition,AlanV.Oppenheim,RonaldW.Schafer.
Copyright©2010byPearsonEducation,Inc. PublishedbyPearsonPrenticeHall.Allrightsreserved.
1
Introduction
on Moore’s Law, the processing capability of both special-purpose signal processing
microprocessorsandpersonalcomputersislikelytoincreasebyseveralordersofmag-
nitudeoverthenext10years.Clearly,theimportanceandroleofsignalprocessingwill
continuetoexpandatanacceleratingratewellintothefuture.
Signalprocessingdealswiththerepresentation,transformation,andmanipulation
ofsignalsandtheinformationthesignalscontain.Forexample,wemaywishtosepa-
ratetwoormoresignalsthathavebeencombinedbysomeoperation,suchasaddition,
multiplication,orconvolution,orwemaywanttoenhancesomesignalcomponentor
estimatesomeparameterofasignalmodel.Incommunicationssystems,itisgenerally
necessarytodopreprocessingsuchasmodulation,signalconditioning,andcompression
prior to transmission over a communications channel,and then to carry out postpro-
cessing at the receiver to recover a facsimile of the original signal. Prior to the 1960s,
thetechnologyforsuchsignalprocessingwasalmostexclusivelycontinuous-timeana-
logtechnology.2Acontinualandmajorshifttodigitaltechnologieshasresultedfrom
the rapid evolution of digital computers and microprocessors and low-cost chips for
analog to digital (A/D) and digital to analog (D/A) conversion.These developments
intechnologyhavebeenreinforcedbymanyimportanttheoreticaldevelopments,such
as the fast Fourier transform (FFT) algorithm,parametric signal modeling,multirate
techniques,polyphasefilterimplementation,andnewwaysofrepresentingsignals,such
aswithwaveletexpansions.Asjustoneexampleofthisshift,analogradiocommunica-
tionsystemsareevolvingintoreconfigurable“softwareradios”thatareimplemented
almostexclusivelywithdigitalcomputation.
Discrete-timesignalprocessingisbasedonprocessingofnumericsequencesin-
dexed on integer variables rather than functions of a continuous independent vari-
able.Indigitalsignalprocessing(DSP),signalsarerepresentedbysequencesoffinite-
precisionnumbers,andprocessingisimplementedusingdigitalcomputation.Themore
generaltermdiscrete-timesignalprocessingincludesdigitalsignalprocessingasaspe-
cial case but also includes the possibility that sequences of samples (sampled data)
could be processed with other discrete-time technologies. Often the distinction be-
tweenthetermsdiscrete-timesignalprocessinganddigitalsignalprocessingisofminor
importance, since both are concerned with discrete-time signals. This is particularly
truewhenhigh-precisioncomputationisemployed.Althoughtherearemanyexamples
inwhichsignalstobeprocessedareinherentlydiscrete-timesequences,mostapplica-
tions involve the use of discrete-time technology for processing signals that originate
ascontinuous-timesignals.Inthiscase,acontinuous-timesignalistypicallyconverted
intoasequenceofsamples,i.e.,adiscrete-timesignal.Indeed,oneofthemostimpor-
tantspurstowidespreadapplicationofdigitalsignalprocessingwasthedevelopment
of low-costA/D,D/A conversion chips based on differential quantization with noise
shaping. After discrete-time processing, the output sequence is converted back to a
continuous-timesignal.Real-timeoperationisoftenrequiredordesirableforsuchsys-
tems.Ascomputerspeedshaveincreased,discrete-timeprocessingofcontinuous-time
signals in real time has become commonplace in communication systems, radar and
sonar,speech and video coding and enhancement,biomedical engineering,and many
2Inageneralcontext,weshallrefertotheindependentvariableas“time,”eventhoughinspecific
contexts,theindependentvariablemaytakeonanyofabroadrangeofpossibledimensions.Consequently,
continuoustimeanddiscretetimeshouldbethoughtofasgenerictermsreferringtoacontinuousindependent
variableandadiscreteindependentvariable,respectively.
2
Introduction
otherareasofapplication.Non-real-timeapplicationsarealsocommon.Thecompact
discplayerandMP3playerareexamplesofasymmetricsystemsinwhichaninputsignal
isprocessedonlyonce.Theinitialprocessingmayoccurinrealtime,slowerthanreal
time,or even faster than real time.The processed form of the input is stored (on the
compact disc or in a solid state memory),and final processing for reconstructing the
audiosignaliscarriedoutinrealtimewhentheoutputisplayedbackforlistening.The
compactdiscandMP3recordingandplaybacksystemsrelyonmanysignalprocessing
concepts.
FinancialEngineeringrepresentsanotherrapidlyemergingfieldwhichincorpo-
rates many signal processing concepts and techniques. Effective modeling,prediction
andfilteringofeconomicdatacanresultinsignificantgainsineconomicperformance
and stability. Portfolio investment managers,for example,are relying increasingly on
using sophisticated signal processing since even a very small increase in signal pre-
dictabilityorsignal-to-noiseratio(SNR)canresultinsignificantgaininperformance.
Anotherimportantareaofsignalprocessingissignalinterpretation.Insuchcon-
texts,theobjectiveoftheprocessingistoobtainacharacterizationoftheinputsignal.
For example,in a speech recognition or understanding system,the objective is to in-
terpret the input signal or extract information from it. Typically, such a system will
applydigitalpre-processing(filtering,parameterestimation,andsoon)followedbya
patternrecognitionsystemtoproduceasymbolicrepresentation,suchasaphonemic
transcription of the speech.This symbolic output can,in turn,be the input to a sym-
bolicprocessingsystem,suchasarules-basedexpertsystem,toprovidethefinalsignal
interpretation.
Still another relatively new category of signal processing involves the symbolic
manipulation of signal processing expressions. This type of processing is potentially
useful in signal processing workstations and for the computer-aided design of signal
processingsystems.Inthisclassofprocessing,signalsandsystemsarerepresentedand
manipulatedasabstractdataobjects.Object-orientedprogramminglanguagesprovide
aconvenientenvironmentformanipulatingsignals,systems,andsignalprocessingex-
pressionswithoutexplicitlyevaluatingthedatasequences.Thesophisticationofsystems
designedtodosignalexpressionprocessingisdirectlyinfluencedbytheincorporation
offundamentalsignalprocessingconcepts,theorems,andproperties,suchasthosethat
formthebasisforthisbook.Forexample,asignalprocessingenvironmentthatincor-
poratesthepropertythatconvolutioninthetimedomaincorrespondstomultiplication
inthefrequencydomaincanexploreavarietyofrearrangementsoffilteringstructures,
includingthoseinvolvingthedirectuseofthediscreteFouriertransform(DFT)andthe
FFTalgorithm.Similarly,environmentsthatincorporatetherelationshipbetweensam-
plingrateandaliasingcanmakeeffectiveuseofdecimationandinterpolationstrategies
forfilterimplementation.Similarideasarecurrentlybeingexploredforimplementing
signal processing in network environments. In this type of environment,data can po-
tentiallybetaggedwithahigh-leveldescriptionoftheprocessingtobedone,andthe
detailsoftheimplementationcanbebaseddynamicallyontheresourcesavailableon
thenetwork.
Many concepts and design techniques are now incorporated into the structure
ofsophisticatedsoftwaresystemssuchasMATLAB,Simulink,Mathematica,andLab-
VIEW.Inmanycaseswherediscrete-timesignalsareacquiredandstoredincomputers,
these tools allow extremely sophisticated signal processing operations to be formed
3
Introduction
from basic functions. In such cases, it is not generally necessary to know the details
oftheunderlyingalgorithmthatimplementsthecomputationofanoperationlikethe
FFT,butneverthelessitisessentialtounderstandwhatiscomputedandhowitshould
beinterpreted.Inotherwords,agoodunderstandingoftheconceptsconsideredinthis
textisessentialforintelligentuseofthesignalprocessingsoftwaretoolsthatarenow
widelyavailable.
Signalprocessingproblemsarenotconfined,ofcourse,toone-dimensionalsignals.
Althoughtherearesomefundamentaldifferencesinthetheoriesforone-dimensional
and multidimensional signal processing,much of the material that we discuss in this
text has a direct counterpart in multidimensional systems.The theory of multidimen-
sionaldigitalsignalprocessingispresentedindetailinavarietyofreferencesincluding
DudgeonandMersereau(1984),Lim(1989),andBracewell(1994).3 Manyimagepro-
cessing applications require the use of two-dimensional signal processing techniques.
Thisisthecaseinsuchareasasvideocoding,medicalimaging,enhancementandanaly-
sisofaerialphotographs,analysisofsatelliteweatherphotos,andenhancementofvideo
transmissionsfromlunaranddeep-spaceprobes.Applicationsofmultidimensionaldig-
italsignalprocessingtoimageprocessingarediscussed,forexample,inMacovski(1983),
Castleman(1996),Jain(1989),Bovic(ed.)(2005),Woods(2006),GonzalezandWoods
(2007),andPratt(2007).Seismicdataanalysisasrequiredinoilexploration,earthquake
measurement,and nuclear test monitoring also uses multidimensional signal process-
ingtechniques.Seismicapplicationsarediscussedin,forexample,RobinsonandTreitel
(1980)andRobinsonandDurrani(1985).
Multidimensionalsignalprocessingisonlyoneofmanyadvancedandspecialized
topicsthatbuildonsignal-processingfundamentals.Spectralanalysisbasedontheuse
of the DFT and the use of signal modeling is another particularly rich and important
aspectofsignalprocessing.Highresolutionspectrumanalysismethodsalsoarebased
onrepresentingthesignaltobeanalyzedastheresponseofadiscrete-timelineartime-
invariant(LTI)filtertoeitheranimpulseortowhitenoise.Spectralanalysisisachieved
byestimatingtheparameters(e.g.,thedifferenceequationcoefficients)ofthesystem
and then evaluating the magnitude squared of the frequency response of the model
filter.DetaileddiscussionsofspectrumanalysiscanbefoundinthetextsbyKay(1988),
Marple(1987),Therrien(1992),Hayes(1996)andStoicaandMoses(2005).
Signal modeling also plays an important role in data compression and coding,
andhereagain,thefundamentalsofdifferenceequationsprovidethebasisforunder-
standingmanyofthesetechniques.Forexample,oneclassofsignalcodingtechniques,
referredtoaslinearpredictivecoding(LPC),exploitsthenotionthatifasignalisthe
responseofacertainclassofdiscrete-timefilters,thesignalvalueatanytimeindexisa
linearfunctionof(andthuslinearlypredictablefrom)previousvalues.Consequently,
efficientsignalrepresentationscanbeobtainedbyestimatingthesepredictionparam-
etersandusingthemalongwiththepredictionerrortorepresentthesignal.Thesignal
canthenberegeneratedwhenneededfromthemodelparameters.Thisclassofsignal
3AuthorsnamesanddatesareusedtorefertobooksandpaperslistedintheBibliographyattheend
ofthischapter.
4
Introduction
codingtechniqueshasbeenparticularlyeffectiveinspeechcodingandisdescribedin
considerable detail in Jayant and Noll (1984),Markel and Gray (1976),Rabiner and
Schafer(1978)andQuatieri(2002).
Anotheradvancedtopicofconsiderableimportanceisadaptivesignalprocessing.
Adaptivesystemsrepresentaparticularclassoftime-varyingand,insomesense,non-
linearsystemswithbroadapplicationandwithestablishedandeffectivetechniquesfor
theirdesignandanalysis.Again,manyofthesetechniquesbuildfromthefundamentals
of discrete-time signal processing. Details of adaptive signal processing are given by
WidrowandStearns(1985),Haykin(2002)andSayed(2008).
These represent only a few of the many advanced topics that extend from the
contentcoveredinthistext.Othersincludeadvancedandspecializedfilterdesignpro-
cedures,avarietyofspecializedalgorithmsforevaluationoftheFouriertransform,spe-
cializedfilterstructures,andvariousadvancedmultiratesignalprocessingtechniques,
including wavelet transforms. (See Burrus,Gopinath,and Guo (1997),Vaidyanathan
(1993)andVetterliandKovacˇevic´ (1995)forintroductionstothesetopics.)
It has often been said that the purpose of a fundamental textbook should be to
uncover,ratherthancover,asubject.Wehavebeenguidedbythisphilosophy.Thereis
arichvarietyofbothchallengingtheoryandcompellingapplicationstobeuncovered
bythosewhodiligentlypreparethemselveswithastudyofthefundamentalsofDSP.
HISTORIC PERSPECTIVE
Discrete-timesignalprocessinghasadvancedinunevenstepsovertime.Lookingback
at the development of the field of discrete-time signal processing provides a valuable
perspective on fundamentals that will remain central to the field for a long time to
come.Sincetheinventionofcalculusinthe17th century,scientistsandengineershave
developedmodelstorepresentphysicalphenomenaintermsoffunctionsofcontinuous
variablesanddifferentialequations.However,numerictechniqueshavebeenusedto
solvetheseequationswhenanalyticalsolutionsarenotpossible.Indeed,Newtonused
finite-differencemethodsthatarespecialcasesofsomeofthediscrete-timesystemsthat
wepresentinthistext.Mathematiciansofthe18thcentury,suchasEuler,Bernoulli,and
Lagrange,developedmethodsfornumericintegrationandinterpolationoffunctionsof
acontinuousvariable.InterestinghistoricresearchbyHeideman,Johnson,andBurrus
(1984)showedthatGaussdiscoveredthefundamentalprincipleoftheFFTasearlyas
1805—evenbeforethepublicationofFourier’streatiseonharmonicseriesrepresenta-
tionoffunctions.
Untiltheearly1950s,signalprocessingaswehavedefineditwastypicallycarried
outwithanalogsystemsimplementedwithelectroniccircuitsorevenwithmechanical
devices.Eventhoughdigitalcomputerswerebecomingavailableinbusinessenviron-
ments and in scientific laboratories, they were expensive and had relatively limited
capabilities.Aboutthattime,theneedformoresophisticatedsignalprocessinginsome
applicationareascreatedconsiderableinterestindiscrete-timesignalprocessing.One
ofthefirstusesofdigitalcomputersinDSPwasingeophysicalexploration,whererel-
ativelylowfrequencyseismicsignalscouldbedigitizedandrecordedonmagnetictape
5
