Signals and systems

453 Pages·2004·4.942 MB·English

by  Chen, Chi-Tsong

Chen-42037 book January13,2004 14:22
SIGNALS AND SYSTEMS
THIRD EDITION
Chi-Tsong Chen
StateUniversityofNewYorkatStonyBrook
New York Oxford
OXFORD UNIVERSITY PRESS
2004
Chen-42037 book January13,2004 14:22
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Rev.ed.of:Systemandsignalanalysis.2nded.c1994.
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1.Systemanalysis.2.Signaltheory(Telecommunication) I.Chen,Chi-Tsong.System
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PREFACE
Thistextintroducesbasicconceptsinsignalsandsystemsandtheirassociatedmathematicaland
computationaltools.Itspurposeistoprovideacommonbackgroundforsubsequentcoursesin
electricalengineering:communications,control,electroniccircuits,filterdesign,anddigitalsig-
nalprocessing.Itisalsousefulinthestudyofvibrationsandcontrolinmechanicalengineering.
The reader is assumed to have knowledge of general physics (including simple circuit analy-
sis),simplematrixoperations,andbasiccalculus.Someknowledgeofdifferentialequationsis
helpfulbutnotessential.
This text is a revision of System and Signal Analysis, 2nd edition, which was published in
1994. It differs considerably in presentation and emphasis from the second edition and from
othertextsonsignalsandsystems:
1. Most texts on signals and systems prepare students for subsequent courses in commu-
nications, control, and signal processing. Some lean more heavily on communications,
some more on control. The course on signals and systems at Stony Brook was changed
in the mid-1990s from an elective to a required course for all students in the electrical
engineeringandcomputerengineeringprograms.Toreflectthischange,thistextuses(in
additiontopassiveRLCnetworksandsimplemechanicalsystems)operationalamplifiers
(opamps)asexamplestodiscusssystemconcepts,introducesop-ampcircuitimplementa-
tion,anddiscussesfeedbackdesign.Thus,thistextalsopreparesstudentsforsubsequent
electronicscourses.
2. Most texts on signals and systems give a fairly complete treatment of transform theory
and then discuss engineering applications. This revision, in response to feedback from
alumniworkinginindustry,introducesengineeringconceptsattheearliestpossiblestage
and then introduces transform theory only as needed. Thus, the discussion of transform
theoryisnotexhaustive.Weskipmanytopicsthatareonlyofacademicinterestorrarely
ariseinpractice.Instead,wefocusonconcepts(suchasspeedsofresponseandoperational
frequencyrangesofdevices)thatarerelevanttoengineering.Wealsoincludeengineering
constraints when discussing mathematics, such as in the feedback implementation of
inverse systems. Otherwise, a design may become simply a mathematical exercise with
noengineeringbearing.
3. Thisrevisionstressescomputercomputation.ItintroducesthefastFouriertransform(FFT)
to compute frequency spectra. We discuss its actual employment rather than its internal
xiii
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xiv PREFACE
programming.Becausemost,ifnotall,systemresponsesinMATLAB1arecomputedfrom
state-space(ss)equations,wedevelopssequationsanddiscusstheircomputercomputation
andtheirop-ampcircuitimplementations.
4. Mathematicaltrainingisimportantinengineering,webelieve,notinitsabstractconcepts
andtheoreticalresultsbutinitsrigorousmethodology.Thus,wedefineeverytermcarefully
and develop every topic logically. For example, we define λ as a zero of the rational
function H(s)= N(s)/D(s)if H(λ)=0.Manyengineeringtextsdefineλasazeroof
H(s)if N(λ)=0.Thisiscorrectonlyif D(λ)=(cid:2) 0or N(s)and D(s)havenocommon
factor at λ. This leads to the concepts of coprimeness and degree. These concepts are
important in engineering, because they are used in minimal realizations and in optimal
andpole-placementdesignsincontrol,yettheyarenotdiscussedinmostothertexts.We
also distinguish between magnitudes and amplitudes and between lumped systems and
distributedsystems.Notethattheresultsinlumpedsystemsarenotnecessarilyapplicable
todistributedsystems.Foreverypositiveresult,weshowanegativeresulttoillustratethe
importanceoftheconditioninvolved.
5. This revision tries to present every topic in the simplest possible way. For example, the
discussionleadingtotheFFTisself-containedevenwithoutdevelopingthediscreteFourier
transform(DFT).ThepresentationoftheRouthtestforcheckingstabilityisbelievedto
besimplerthantheconventionalcross-productmethod.Wealsosimplifytheconceptsof
controllabilityandobservabilityinssequationstotheconceptsofcoprimenessanddegree
intransferfunctionsandusethemtodiscusstheredundancyofsystems.
The following is a brief description of each chapter. It also provides a global view and
introduceskeyterminologyofthesubjectarea:
1. Signals:Weintroducecontinuous-time(CT),discrete-time(DT),anddigitalsignals.We
give reasons for not studying digital signals even though they are used in all computer
computationanddigitalsignalprocessing.WethenintroducesomesimpleCTsignals,dis-
cusstheirvariousmanipulations,andgivereasonsforusingcomplexexponentialfunctions
todefinefrequencycomponentsofCTsinusoids.UnlikeCTsignals,DTsignalscanbe
plottedagainsttimeinstantsortimeindex(withoutspecifyingthesamplingperiod).The
latterisusedintheirmanipulations,2andtheformerisusedindiscussingtheirfrequency
content.WeshowthatthefrequencyofDTsinusoidscannotbedirectlydefined.Wethen
defineitformallyfromthefrequencyofCTsinusoidsandjustifyitphysically.Thisleads
totheconceptsofNyquistfrequencyrangeforDTsignalsandfrequencyaliasingdueto
timesampling.
2. Systems:Wemodelasystemasablackboxwhoseeveryinput,asignal,excitesaunique
output, another signal. We then introduce systems with and without memory. For sys-
temswithmemory,weintroducetheconceptsofcausality,state(setofinitialconditions),
lumpedness, zero-state (forced) responses, and zero-input (natural) responses. Wethen
1MATLABisaregisteredtrademarkofTheMathWorks,Inc.
2Theyinvolvemanipulationsofstreamsofnumbers.Theproceduresarethesameforanysamplingperiod
T and,consequently,areindependentofT.
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PREFACE xv
introducetheconceptsoflinearity(L)andtimeinvariance(TI).Wediscusstheirimplica-
tionsandexplainwhythestudyofLTIsystemsisrelativelysimple.Wethenuseexamples,
includingopamps,toshowthatLTIlumpedsystemsareobtainedinpracticebymodeling,
approximation,andsimplificationandarevalidonlyforlimitedinputsignalranges.
3. Mathematicaldescriptionsofsystems:Weusetheconceptsoflinearityandtimeinvariance
todevelopdiscreteandintegralconvolutionsforLTIsystemsanddifferenceanddifferential
equationsforLTIandlumpedsystems.Thedevelopmentisgenericandisapplicableto
anysystem,beitelectrical,mechanical,chemical,orbiomedical.
4. CTsignalanalysis:WeusetheFourierseriesandFouriertransformtodevelopfrequency
spectra for CT signals. We discuss several sufficient conditions for signals to have fre-
quencyspectraandarguethatfrequencyspectraofmostpracticalsignalsarewell-defined,
bounded,andcontinuous.Weshowthatfrequencyspectraofsignalsrevealexplicitlythe
distributionofenergyinfrequencies.Theconceptisessentialindiscussingmanytopics
inlaterchapters.
5. The sampling theorem and spectral computation: Frequency spectra of most, if not all,
practicalsignalscannotbeexpressedinclosedformandcannotbecomputedanalytically.
Theonlywaytocomputethemisnumericallyfromtheirtimesamples.Thus,weestablish
the relationship between the frequency spectra of a CT signal and its sampled DT se-
quenceandthenpresenttheNyquistsamplingtheorem.Wethenintroducethe fastFourier
transform(FFT)tocomputefrequencyspectraofDTandCTsignals.
6. CTsystemqualitativeanalysis:WeintroducetransferfunctionsthroughtheLaplacetrans-
formandgivereasonsforstudyingonlyproperrationaltransferfunctions.Weintroduce
polesandzerosandusethemtodevelopgeneralresponsesofsystems.Wethenintroduce
theconceptsoffrequencyresponseandstability3forsystemsandestablishtheequation
Y(jω)= H(jω)U(jω) (P.1)
whereY(jω)istheoutput’sfrequencyspectrum,U(jω)istheinput’sfrequencyspectrum,
and H(jω) is a system’s frequency response. We show that the equation has physical
meaning only if the system is stable. We discuss how to compute the Fourier transform
usingtheLaplacetransformandshowthatthephasoranalysisdiscussedinmostnetwork
textsisapplicableonlytostablesystems.Wegivereasonsfornotdiscussinginthistext
the Fourier analysis of systems even though it is discussed in most other texts. We also
givereasonsfornotusingtransferfunctionsincomputercomputation.
7. CTsystemquantitativeanalysis:Forcomputercomputation,wetransformtransferfunc-
tionsintostate-space(ss)equations,calledtherealizationproblem.Thename“realization”
isjustifiedbythefactthateveryssequationcanbereadilysimulatedonacomputerand
implementedusinganop-ampcircuit.Thusssequationsaremoreconvenientforcomputer
computationandsynthesis,whereastransferfunctionsaremoreconvenientforqualitative
3EverypassiveRLCnetworkisstable,anditsstabilitystudyisunnecessary.However,acircuitthatcontains
activeelementssuchasopampsoracomputerprogramcaneasilybecomeunstable.Thusitsstability
studyisimperative.
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xvi PREFACE
analysisanddesign.Theanalyticalstudyofssequationsisnotdiscussedbecauseitplays
noroleintheapplicationsmentionedabove.Wealsocomparethetwodescriptionsand
justifytheuseoftransferfunctionsindisregardingzero-inputresponsesofsystems.We
thendiscussanidentificationschemetodevelopamorerealisticmodelforopampsand
cautiontheuseoflinearsweepsinusoidsinidentification.
8. Applications:Thischapterintroducesthreeindependenttopics.Thefirsttopicisbasicin
allengineeringdisciplines,thesecondtopicisbasicincontrolandelectronics,andthe
lasttopicisbasicincommunication:
(a) Model reductions: Amplifiers, seismometers, and accelerometers are all based on
reducedmodels.Thus,modelreductioniswidelyusedinpractice.Weuse(P.1)to
developoperationalfrequencyrangesfordevicesanddemonstratethatadevicewill
yieldtheintendedresultonlyifthefrequencyspectraofinputsignalslieinsidethe
operationalfrequencyrangeofthedevice.Notethatseismometersandaccelerom-
eters have transfer functions of the same form, but they have different operational
frequencyrangesand,consequently,differentreducedmodelsanddifferentdesign
requirements.Wealsoshowthattheaccelerometersusedinautomobilestotrigger
airbagsaremuchmorecomplexthanthesimplemodeldiscussedinthisoranyother
similar text and cannot be easily analyzed. Thus engineering is more than math-
ematics and physics; it needs innovation, the construction of prototypes and their
repetitivetestingsandimprovements,andyearsofdevelopment.
(b) Feedback:Wediscusstheloadingprobleminconnectingtwosystems.Wethenuse
an example to demonstrate the main reason for using feedback: reduction of the
effectsofparametervariations.Feedback,however,introducesthestabilityproblem.
Weshowthatthestabilityofanegativeorpositivefeedbacksystemisindependent
ofthestabilityofitssubsystems.Wealsousefeedbacktoimplementinversesystems
anddiscussitslimitations.WethendesignWien-bridgeoscillators—directlyandby
usingafeedbackmodel—andrelatetheBarkhausencriteriontothepolecondition.
(c) Modulations:Weintroducetwomodulationschemesandshowtheroleof(P.1)in
theirdemodulations.
9. DT system qualitative analysis: We introduce the z-transform, DT transfer functions,
stability,andfrequencyresponses.
10. DT system quantitative analysis: We develop DT state-space equations directly from
high-orderdifferenceequations.TheprocedurehasnoCTcounterpart.Theremainderof
thischaptercloselyfollowstheCTcase.
MostresultsinthistextcanbeobtainedbytypingasmallnumberofMATLABfunctions.4
Thus,itismoreimportantthanevertounderstandbasicideasandprocedures.Mostexercisesin
thetextaredesignedtocheckunderstandingofthetopicinvolvedandrequireminimalnumerical
computation. Thus the reader should solve the exercises before proceeding to the next topic.
Wealsosuggestthatthereadersolvetheproblemsattheendofeachchapterbyhandandthen
4AllresultsinthistextareobtainedusingMATLAB5.3StudentVersion.
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PREFACE xvii
verifytheresultsusingacomputer.Inaddition,werecommendthatthereaderrepeatallofthe
programsinthetext.Theprogramswillyieldonlyessentialresultsbecausetheprogramsskip
nonessentialfunctionssuchasthesizingoffiguresandthedrawingofcoordinates.
Thelogicalsequenceofthechaptersisasfollows:

Sections5.1–5.3CSe⇒hcatpiSoteenrcst7i8o.n1s–85..34–5.6
(cid:5)
Chapters1–4⇒ Chapter6⇒ Section8.4
 SSeeccttiioonn88..75–8.8⇒Section8.6
Chapter9⇒Chapter10
Thistextcontainsmorematerialthancanbecoveredinonesemester.Aone-semestersopho-
more/juniorcourseatStonyBrookcovers(skippingtheasteriskedsections)Chapters1through4,
Sections5.1through5.3,Chapters6and7,andpartsofChapter8.Clearly,otherarrangements
arealsopossible.Asolutionsmanualisavailablefromthepublisher.
Many people helped me in writing this text. Mr. Anthony Olivo performed many op-amp
circuitexperimentsforme.IconsultedProfessorsArmenZemanianandJohnMurraywhenever
I had any questions or doubts. The first draft of this revision was reviewed by a number of
reviewers.Theircommentspromptedmetorearrangeandrewritemanysections.Manypeople
atOxfordUniversityPress,includingPeterC.Gordon,DanielleChristensen,BarbaraBrown,
TrentHaywood,MaryHopkins,andBrianKinsey,weremosthelpfulinthisundertaking.Ithank
themall.Finally,Ithankmywife,Bih-Jau,forhersupport.
A NOTE TO THE READER
WhenIwasanundergraduatestudentaboutforty-fiveyearsago,Idideverysingleoneofthe
assigned problems and was an “A” student. I believed that I understood most subjects well.
This belief was reinforced by my passing a competitive entrance exam to a master’s-degree
programinTaiwan.AgainIcompletedthedegreewithflyingcolorsandwasconfidentformy
nextchallenge.
MyconfidencewascompletelyshatteredwhenIstartedtodoresearchunderProfessorCharles
A.DesoerattheUniversityofCalifornia,Berkeley.Underhiscriticalandconstantquestioning,
IrealizedthatIdidnotunderstandthesubjectofstudyatall.Moreimportant,Idiscoveredthat
my method of studying had been incorrect: learning only the mechanics of solving problems
withoutlearningtheunderlyingconcepts.Fromthattimeon,wheneverIstudiedatopic,Iwould
pondereverystatementcarefullyandthenstudyitsimplications.Aretheimplicationsstillvalid
ifsomewordinthestatementismissing?Why?Aftersomethought,Iwouldre-readthetopic
orarticle.Itoftentookmeseveraliterationsofponderingandre-readingtofullygraspcertain
ideasandresults.Ialsolearnedtoconstructsimpleexamplestogaininsightand,bykeepingin
mindthegoalsofastudy,todifferentiatebetweenwhatisessentialandwhatissecondaryor
notimportant.Ittakesagreatdealoftimeandthoughttoreallyunderstandasubject.
Chen-42037 book January13,2004 14:22
xviii PREFACE
AsaconsequenceofmyPh.D.training,Ibecamefascinatedbymathematicsforitsabsolute
correctnessandrigorousdevelopment—noambiguityandnoapproximation.Thusintheearly
partofmyteachingcareer,Itendedtoteachmoremathematicsthanengineering,asisevident
inthefactthatIexpandedthebookLinearSystemTheoryandDesignfrom431to662pages
betweenitsfirstandsecondeditions.Withtheinformationexplosion,awideninggapbetween
theory and practice, less-prepared students, and our limited time and energy, I realized in the
middleofmyteachingcareerthatitisunnecessarytoburdenengineeringstudents,especially
non-Ph.D.students,withtoomuchmathematics.ThusIcutthethirdeditionoftheaforemen-
tionedbookinhalf,from662to334pages,bydeletingthosetopicsthatweremostlyofacademic
interestandseemedtohavenopracticalapplication.Inthesamespirit,myintentionwithSignals
andSystems,3rdedition,hasbeentodevelopafocusedandconcisetextonsignalsandsystems
that still contains all material that an undergraduate student or a practicing engineer needs to
know.IhopeIhavesucceededinthisendeavor.
Studentstakingacourseonsignalsandsystemsusuallytakethreeorfourothercoursesatthe
sametime.Theymayalsohavemanydistractions:part-timejobs,relationships,ortheInternet.
Theysimplydonothavethetimetoreallyponderatopic.ThusIfullysympathizewiththeir
lackofunderstanding.Whenstudentscometomyofficetoaskquestions,Ialwaysinsistthat
theytrytosolvetheproblemsthemselvesbygoingbacktotheoriginaldefinitionsandthenby
developingtheanswersstepbystep.Mostofthetime,thestudentsdiscoverthatthequestions
were not difficult at all. Thus if the reader finds a topic difficult, he or she should go back to
thinkaboutthebasicdefinitionsandthenfollowthestepslogically.Donotgetdiscouraged.We
suggestthatthereaderreferoccasionallytotheprefacetokeepinmindthegoalsofthestudy.
It is hoped that after finishing this book, the reader will be comfortable with all the italicized
termsinthepreface.
Chi-TsongChen
October2003
Chen-42037 book January13,2004 14:22
CONTENTS
Preface xiii
1 Signals 1
1.1 Introduction 1
1.2 Continuous-Time(CT),Discrete-Time(DT),andDigitalSignals 1
1.3 ElementaryCTSignals 7
1.4 ManipulationsofCTSignals 10
1.4.1 ShiftingandFlipping 10
1.4.2 MultiplicationandAddition 12
1.4.3 Modulation 13
1.4.4 WindowsandPulses 14
1.5 Impulse 16
1.5.1 Piecewise-ConstantApproximationofCTSignals 20
1.6 ElementaryDTSignalsandTheirManipulation 21
1.7 CTSinusoidalSignals 26
1.7.1 FrequencyComponents 27
1.7.2 ComplexExponentials—PositiveandNegativeFrequencies 28
1.7.3 MagnitudesandPhases;EvenandOdd 29
1.8 DTSinusoidalSequencesandNyquistFrequencyRange 34
1.9 SamplingandFrequencyAliasing 40
Problems 44
2 Systems 49
2.1 Introduction 49
2.2 CTSystemswithandwithoutMemory 50
vii