Discrete-Time Signal Processing PDF
Preview Discrete-Time Signal Processing
Discrete-Time Signal Processing Alan V. Oppenheim Ronald W. Schafer Third Edition ISBN 10: 1-292-02572-7 ISBN 13: 978-1-292-02572-8 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. ISBN 10: 1-292-02572-7 ISBN 13: 978-1-292-02572-8 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America 1112356789902106891542381537171771551 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
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