37chapter 1 signals and systems

Chapter1SignalsAndSystemsContents„Descriptionofsignals„Transformationsoftheindependentvariable„Somebasicsignals„Systemsandtheirmathematicalmodels„Basicsystemsproperties1.1Continuous-TimeandDiscrete-TimeSignals1.1.1ExamplesandMathematicalRepresentation(1)AsimpleRCcircuitSourcevoltageVsandCapacitorvoltageVcA.Examples(2)AnautomobileForceffromengineRetardingfrictionalforceρVVelocityV(3)ASpeechSignal(4)APicture(5)vitalstatistics(人口统计）Note„Inthisbook,wefocusonourattentiononsignalsinvolvingasingleindependentvariable.„Forconvenience,wewillgenerallyrefertotheindependentvariableastime,althoughitmaynotinfactrepresenttimeinspecificapplications.0tA)(tx10t)(tx⎪⎪⎩⎪⎪⎨⎧≥−=0,00,)(ttatetxRttAtx∈+=),sin()(ϕωB.Twobasictypesofsignalst:continuoustimex(t):continuumofvalue1.Continuous-Timesignal2.Discrete-Timesignaln:discretetimex[n]:consequenceseries199020028001600Example1:1990-2002年的某村农民的年平均收入0Δ124.08.0Δ1Δ2Δ3t0124.08.0123n4567111098)(tx][nxsk81=ΔSamplingExample2:x[n]issampledfromx(t)(1)FunctionRepresentationExample:x(t)=cosω0tx(t)=ejω0t(2)GraphicalRepresentationExample:(Seepagebefore)(3)Sequence-representationfordiscrete-timesignals:x[n]={-21321–1}orx[n]=(-21321–1)C.Representation3−1.1.2SignalEnergyandPowerInstantaneouspower:)()(1)()()(22tiRtvRtitvtp⋅==⋅=LetR=1Ω,so)()()()(222txtvtitp===+R_)(tv)(tiEnergy:t1≤t≤t2∫∫∫==212121)()()(22ttttttdttxdttvdttpAveragePower:∫∫−=−2121)(1)(121212ttttdttxttdttpttTotalEnergyAveragePower∑=+−21212][11nnnnxnnDefinition:∫212)(ttdttx∫−21212)(1ttdttxtt∑=212][nnnnxDiscrete-Time:(n1≤n≤n2)Wewillfrequentlyfinditconvenienttoconsidersignalsthattakeoncomplexvalues.Continuous-Time:(t1≤t≤t2)when∑−=∞→∞+=NNnNnxNP2][121lim∫∫∞∞−−∞→∞==dttxdttxETTT22)()(lim∞∞−∞∞−ntTotalEnergyAveragePower∑∑∞−∞=−=∞→∞==nNNnNnxnxE22][][lim∫−∞→∞=TTTdttxTP2)(21limNote:™Itisimportanttorememberthattheterms“Power”and“energy”areusedhereindependentlyofthequantitiesactuallyarerelatedtophysicalenergy.™Withthesedefinitions,wecanidentifythreeimportantclassofsignals:a.finitetotalenergyb.finiteaveragepowerc.infinitetotalenergy,infiniteaveragepower∞∞E02lim==∞∞→∞TEPT∞∞P),0(lim∞==∫−∞∞→∞∞TTTPEthenPif∞=∞=∞∞PE,1.2TransformationoftheIndependentVariable1.2.1ExamplesofTransformations1.TimeShiftx(t-t0),x[n-n0]t00Advancen00Delay2.TimeReversalx(-t),x[-n]——Reflectionofx(t)orx[n]3.TimeScalingx(at)(a0)Stretchifa1Compressedifa1Note:Generally,TimescalingonlyforcontinuoustimesignalsThisisalsocalleddecimationofsignals.（信号的抽取）ExampleSolution1：Solution2：()ft12121−0t122−1t0(1)ft+)1(tf−01−122t)31(tf−t0131−232)(tf−t01121−2−t1−12012)1(tf−)3(tf12t031−32)31(tf+012t32－31shiftreversalScalingreversalreversalshiftshiftScalingScalingExamplef(t)Æf(1-3t)1.2.2PeriodicSignalsAperiodicsignalx(t)(orx[n])hasthepropertythatthereisapositivevalueofT(orintegerN)forwhich:x(t)=x(t+T),foralltx[n]=x[n+N],forallnIfasignalisnotperiodic,itiscalledaperiodicsignal.ThefundamentalperiodT0(N0)ofx(t)(x[n])isthesmallestpositivevalueofT(orN)forwhichtheequationholds.x(t)=Cisaperiodicsignal,butitsfundamentalperiodisundefined.ExamplesofperiodicsignalsExamplesofperiodicsignals1.nAnx83sin][π=Itisperiodicsignal.ItsfundamentalperiodisN0=16.2.⎪⎩⎪⎨⎧≥=0,00,cos)(ttttxItisnotperiodic.3.tBtAtx41sin31cos)(+=ππ8,621==TTx(t)isperiodic.Itsperiodisπ24=TThesmallestmultiplesofT1andT2incommon4.tttx2coscos)(+=ππ==21,2TTItisaperiodic,too.ThereisnothesmallestmultiplesofT1andT2incommon5.nnx4cos][=6.x(t)isaperiodic.nnx83cos][π=ItisperiodicwithperiodN=16.-20-15-10-505101520-101-20-15-10-505101520-101-20-15-10-505101520-202cosπtcos2tcosπt+cos2t1.2.3EvenandOddSignalsNote:Anoddsignalmustnecessarilybe0att=0,orn=0.Evensignal:x(-t)=x(t)orx[-n]=x[n]Oddsignal:x(-t)=-x(t)orx[-n]=-x[n]Even-OddDecomposition:Anysignalcanbebrokenintoasumoftwosignals,oneofwhichisevenandanotherisodd.)]()([21)()}({txtxtxtxEve−+==)]()([21)()}({txtxtxtxOdo−−==]}[][{21][]}[{nxnxnxnxEve−+==]}[][{21][]}[{nxnxnxnxOdo−−==or:Exampleoftheeven-odddecompositonExampleoftheeven-odddecompositonHomework：P57--1.61.91.101.21(a)(b)(c)(d)1.22(a)(b)(g)1.231.241.3ExponentialandSinusoidalSignals1.3.1Continuous-timeComplexExponentialandSinusoidalSignalsThecontinuous-timecomplexexponentialsignalisoftheformatCetx=)(whereCandaare,ingeneral,complexnumbers.Dependinguponthevaluesoftheseparameters,thecomplexexponentialcanexhibitseveraldifferentcharacteristics.A.RealExponentialSignalsx(t)=Ceat(C,aarerealvalue)a0a0growingdecayingB.PeriodicComplexExponentialandSinusoidalSignalsx(t)=ejω0tx(t)isperiodicforx(t)=x(t+T),anditsfundamentalperiodis.x(t)=Ceat,C=1,a=jω0(purelyimaginary)002wTπ=(1)Forejω0t00200001TdtdteETTtjwperiod=⋅==∫∫110==periodperiodETPifω0=0,x(t)=1,thenitisperiodicforanyT0.(2)x(t)=Acos(ω0t+φ)002fwπ=ω0－rad/sf0－HzEuler’sRelation:ejω0t=cosω0t+sinω0tandcosω0t=(ejω0t+e-jω0t)/2sinω0t=(ejω0t-e-jω0t)/2WehavetjjtjjeeAeeAtA0022)cos(0ϖφϖφφϖ−−+=+}Im{)sin(}Re{)cos()(0)(000φφφφ++=+=+twjtwjeAtwAeAtwAifcisacomplexnumber,Re{c}denotesitsrealpart;Im{c}denotestheimaginarypart.-20-15-10-505101520-101-20-15-10-505101520-101-20-15-10-505101520-101ω1ω2ω3ω2ω1ω3y=cos2ty=cos5ty=cos10t(3)Harmonically