通信原理(英文版).ppt

1 Chapter2Signals 2 1ClassificationofSignals2 1 1DeterministicsignalsandrandomsignalsWhatisdeterministicsignal Whatisrandomsignal 2 1 2EnergysignalsandpowersignalsSignalpower LetR 1 thenP V2 R I2R V2 I2Signalenergy LetSrepresentVorI ifSvarieswithtime thenScanberewrittenass t Hence thesignalenergyE s2 t dtEnergysignalsatisfiesAveragepower thenP 0forenergysignal Forpowersignal P 0 i e powersignalhasinfiniteduration Energysignalhasfiniteenergy butitsaveragepowerequals0 Powersignalhasfiniteaveragepower butitsenergyequalsinfinity 2 2 2Characteristicsofdeterministicsignals 2 2 1CharacteristicsinfrequencydomainFrequencyspectrumofpowersignal lets t beaperiodicpowersignal itsperiodisT0 thenwehavewhere 0 2 T0 2 f0 C jn 0 isacomplexfunction C jn 0 Cn ej nwhere Cn amplitudeofthecomponentwithfrequencynf0 n phaseofthecomponentwithfrequencynf0Fourierseriesofsignals t 3 Example2 1 Findthespectrumofaperiodicrectangularwave Solution AssumetheperiodofaperiodicrectangularwaveisT thewidthis andtheamplitudeisV thenItsfrequencyspectrumis 4 Frequencyspectrumfigure 5 Example2 2 Findthefrequencyspectrumofasinusoidalwaveafterfull waverectification Solution AssumetheexpressionofthesignalisItsfrequencyspectrum TheFourierseriesofthesignalis 6 FrequencyspectraldensityofenergysignalsLetanenergysignalbes t thenitsfrequencyspectraldensityisTheinverseFouriertransformofS istheoriginalsignal Example2 3 Findthefrequencyspectraldensityofarectangularpulse Solution LettheexpressionoftherectangularpulsebeThenitsfrequencyspectraldensityisitsFouriertransform 7 Example2 4 Findthewaveformandthefrequencyspectraldensityofasamplefunction Solution ThedefinitionofthesamplefunctionisthefrequencyspectraldensitySa t is Fromtheaboveequation weseethatSa isagatefunction Example2 5 Findtheunitimpulsefunctionanditsfrequencyspectraldensity Solution Unitimpulsefunctionisusuallycalleddfunctiond t ItsdefinitionisThefrequencyspectraldensityof t 8 d t anditsfrequencyspectraldensity Physicalmeaningof function Itisapulsewithinfiniteheight infinitesimalwidth andunitarea Sa t hasthefollowingproperty Whenk amplitude andthezero spacingofthewaveform 0 Hence 9 Characterisiticsof t t isanevenfunction t isthederivativeofunitstepfunction DifferencebetweenfrequencyspectraldensityS f ofenergysignalandfrequencyspectrumofperiodicpowersignal S f continuousspectrum C jn 0 discreteUnitofS f V Hz UnitofC jn 0 VAmplitudeofS f atafrequencypoint infinitesimal 10 Example2 6 Findthefrequencyspectraldensityofacosinusoidalwavewithinfinitelength Solution Lettheexpressionofacosinusoidalwavebef t cos 0t thenaccordingtoeq 2 2 10 F canbewrittenasReferencingeq 2 2 19 theaboveequationcanbewrittenas Introducing t theconceptoffrequencyspectraldensitycanbegeneralizedtopowersignal 11 EnergyspectraldensityLettheenergyofanenergysignals t beE thentheenergyofthesignalisdecidedbyIfitsfrequencyspectraldensityisS f thenfromParseval stheoremwehavewhere S f 2iscalledenergyspectraldensity Theaboveequationcanberewrittenas whereG f S f 2 J Hz isenergyspectraldensity PropertyofG f Sinces t isarealfunction S f 2isanevenfunction 12 PowerspectraldensityLetthetruncatedsignalofs t issT t T 2 t T 2 thenTodefinethepowerspectraldensityofthesignalas obtainthesignalpower 13 2 2 2CharacteristicsintimedomainAutocorrelationfunctionDefinitionoftheautocorrelationfunctionforenergysignal Definitionoftheautocorrelationfunctionforpowersignal Characteristics R isonlydependenton butindependentoft When 0 R ofenergysignalequalstheenergyofthesignal andR ofpowersignalequalstheaveragepowerofthesignal 14 Cross correlationfunctionDefinitionofthecross correlationfunctionforenergysignal Definitionofthecross correlationfunctionforpowersignal Characteristics 1 R12 isdependenton andindependentoft 2 Proof Letx t then 15 2 3Characteristicsofrandomsignals 2 3 1ProbabilitydistributionofrandomvariableConceptofrandomvariable IftherandomoutcomeofatrialAisexpressedbyX thenwecallXarandomvariable andletitsvaluebex Forexample thenumberofcallsreceivedwithinagivenperiodoftimeatthetelephoneexchangeisarandomvariable DistributionfunctionofrandomvariableDefinition FX x P X x Characteristics P a X b P X a P X b P a X b P X b P X a P a X b FX b FX a 16 Distributionfunctionofdiscreterandomvariable LetthevaluesofXbe x1 x2 xi xn theirprobabilitiesarerespectivelyp1 p2 pi pn thenP X x1 0 P X xn 1 P X xi P X x1 P X x2 P X xi Characteristics FX 0FX 1Ifx1 x2 thenFX x1 FX x2 monotonicincreasingfunction 17 Distributionfunctionofcontinuousrandomvariable Whenxiscontinuous fromthedefinitionofdistributionfunctionFX x P X x weknowthatFX x isacontinuousmonotonicincreasingfunction 18 2 3 2ProbabilitydensityofrandomvariableProbabilitydensityofcontinuousrandomvariablepX x DefinitionofpX x MeaningofpX x pX x isthederivativeofFX x andistheslopeofthecurveofFX x P a X b canbefoundfrompX x CharacteristicsofpX x pX x 0 19 ProbabilitydensityofdiscreterandomvariableDistributionfunctionofdiscreterandomvariablecanbewrittenas wherepi probabilityofx xiu x unitstepfunctionFindingthederivativesofthetwosidesoftheaboveequation weobtainitsprobabilitydensity Characteristics Whenx xi px x 0Whenx xi px x 20 2 4Examplesoffrequentlyusedrandomvariables RandomvariablewithnormaldistributionDefinition Probabilitydensitywhere 0 a const Probabilitydensitycurve 21 RandomvariablewithuniformdistributionDefinition probabilitydensitywherea bareconstants Probabilitydensitycurve 22 RandomvariablewithRayleighdistributionDefinition Probabilitydensitywherea 0 andisaconstant Probabilitydensitycurve 23 2 5Numericalcharacteristicsofrandomvariable 2 5 1MathematicalexpectationDefinition forcontinuouserandomvariableCharacteristics IfXandYareindependentofeachother andE X andE Y exist then 24 2 5 2VarianceDefinition whereVariancecanberewrittenas Proof Fordiscretevariable Forcontinuousvariable Characteristics D C 0D X C D X D CX C2D X D X Y D X D Y D X1 X2 Xn D X1 D X2 D Xn 25 2 5 3MomentDefinition thek thmomentofarandomvariableXisk thoriginmomentisthemomentwhena 0 k thcentralmomentisthemomentwhen Characteristics Thefirstoriginmomentisthemathematicalexpectation Thesecondcentralmomentisthevariance 26 2 6Randomprocess 2 6 1BasicconceptofrandomprocessX A t ensumbleconsistingofallpossible realizations ofaneventAX Ai t arealizationofeventA itisadeterminedtimefunctionX A tk valueofthefunctionatthegiventimetkDenoteforshort X A t X t X Ai t Xi t 27 Example receivernoiseNumericalcharacteristicsofrandomprocess Statisticalmean Variance Autocorrelationfunction 28 2 6 2StationaryrandomprocessDefinitionofstationaryrandomprocess Arandomprocesswhosestatisticalcharacteristicsisindependentofthetimeoriginiscalledastationaryrandomprocess or strictstationaryrandomprocess Definitionofgeneralizedstationaryrandomprocess Therandomprocesswhosemean varianceandautocorrelationfunctionareindependentofthetimeoriginCharacteristicsofgeneralizedstationaryrandomprocess Astrictstationaryrandomprocessmustbeageneralizedstationaryrandomprocess butageneralizedstationaryrandomprocessisnotalwaysastrictstationaryrandomprocess 29 2 6 3ErgodicitySignificanceofergodicityArealizationofastationaryrandomprocesscangothroughallstatesoftheprocess Characteristicofergodicity timeaveragemaybereplaedbystatisticalmean Forexample StatisticalmeanofergodicprocessmX AutocorrelationfunctionofergodicprocessRX Ifarandomprocesshasergodicity thenitmustbeastrictstationaryrandomprocess However astrictstationaryrandomprocessisnotalwaysergodic 30 ErgodicityofstationarycommunicationsystemIfthesignalandthenoisearebothergodic thenFirstoriginmomentmX E X t D C componentofsignalSquareoffirstoriginmomentmX2 powerofnormalizedD C componentofsignalSecondoriginmomentE X2 t normalizedaveragepowerofsignalSquarerootofsecondoriginmoment E X2 t 1 2 rootmeansquareofsignalcurrentorvoltageSecondcentralmoment X2 normalizedaveragepowerfoA C componentofsignalIfmX mX2 0 then X2 E X2 t Standarddeviation X rootmeansquareofA C componentofsignalIfmX 0 then Xisrootmeansquareofsignal 31 2 6 4AutocorrelationfunctionandpowerspectraldensityofstationaryrandomprocessCharacteristicsofautocorrelationfunction 32 CharacteristicsofpowerspectraldensityReview powerspectraldensityofdeterministicsignalSimilarly powerspectraldensityofstationaryrandomprocessequals Averagepower 33 Relationshipbetweenautocorrelationfunction powerspectraldensityFromwhere Let t t k t t thentheaboveequationcanbereducedtoHence 34 TheaboveequationdemonstratesthatPX f andR areapairofFouriertransform CharacteristicsofPX f PX f 0 andPX f isarealfunction PX f PX f i e PX f isanevenfunction Example2 7 Letabinarysignalbex t asshowninthefigure Itsamplitudeis aor a andthenumberkofitssignchangesintimeintervalTobeysPoissondistribution where isaveragenumberofsignchangesofamplitudeinunittime FinditsautocorrelationfunctionR andpowerspectraldensityP f 35 Solution Itcanbeseenfromtheabovefigure x t x t hasonlytwopossiblevalues a2or a2 Hence equationcanbereducedtoR a2 occurrenceprobabilityofa2 a2 occurrenceprobabilityof a2 where theoccurrenceprobabilitycanbecalculatedaccordingtoPoissondistributionP k Ifthenumberofsignchangesofx t isevenin second then a2occurs ifthenumberofsignchangesofx t isoddin second then a2occurs ThereforeUse insteadofTinPoissondistribution thenobtain 36 The inPoissondistributionisatimeinterval soitshouldbenonnegative Hence whenthevalueof isnegative theaboveequationshouldberewrittenasCombiningtheabovetwoequations finallyobtain TheP f canbeobtainedfromtheFouriertransformoftheR CurvesofP f andR 37 Example2 8 AssumethepowerspectraldensityP f ofarandomprocessisshownasinthefigure FinditsautocorrelationfunctionR Solution P f isknown where Curveofautocorrelationfunction 38 Example2 9 Findtheautocorrelationfunctionandthepowerspectraldensityofwhitenoise Solution WhitenoisehasuniformpowerspectraldensityPn f Pn f n0 2where n0 single sidepowerspectraldensity W Hz Theautocorrelationfunctionofwhitenoisecanbeobtainedfromitspowerspectraldensity Ascanbeseenthesamplesofwhitenoiseatanytwoadjacentinstants i e 0 areuncorrelated Averagepowerofwhitenoise Theaboveequationshowsthattheaveragepowerofwhitenoiseisinfinity 39 Powerspectraldensity autocorrelationfunctionofband limitedwhitenoisePowerspectraldensityofbandlimitedwhitenoise Ifthebandwidthofawhitenoiseinlimitedintheinterval fH fH thenwehavePn f n0 2 fH f fH 0 elseitsautocorrelationfunctinis Curves 40 2 7Gaussianprocess DefinitionOnedimensionalprobabilitydensityofGaussprocess where a E X t mean 2 E X t a 2 variance standarddeviation Gaussprocessisastationaryprocess henceitsprobabilitydensitypX x t1 isindependentoft1i e pX x t1 pX x CurveofpX x 41 StrictdefinitionofGaussianprocess Jointprobabilitydensityofarbitrarydimensionsatisfyingthefollowingcondition where ak mathematicalexpetactionofxk k standarddeviationofxk B determinantofthenormalizedcovariancematrix B jk algebraiccofactorofbjkin B bjk normalizedcovariancefunction i e 42 Characteristicsofn dimensionalGaussprocesspX x1 x2 xn t1 t2 tn isdecidedonlybyai i andbjkofeveryrandomvariables soitisageneralizedstationaryrandomprocess Ifx1 x2 xnareuncorrelatedoneanother thenwhenj k bjk 0 Now i e then dimensionaljointprobabilitydensityequalstheproductofeachonedimensionalprobabilitydensity Ifthecross correlationfunctionoftworandomvariablesequals0 thentheyareuncorrelatedtoeachother ifthetwodimensionaljointprobabilitydensityoftworandomvariablesisequaltotheproductoftheonedimensionalprobabilitydensities thenitisindependentofeachother Twouncorrelatedrandomvariablesarenotalwaysindependentofeachother andtwoindependentrandomvariablesarecertainlyuncorrelated RandomvariablesofGaussianprocessareuncorrelatedandindependentofoneanother 43 Characteristicsofprobabilitydensityofnormaldistributionp x issymmetricaltox a i e p x ismonotonicallyincreasingin a andmonotonicallydecreasingin a andreachesitsmax ata ThemaximumvalueisWhenx orx p x 0 Ifa 0 1 thenthedistributioniscalledstandardnormaldistribution 44 NormaldistributionfunctionTheintegralofnormalprobabilitydensityfunctionisdefinedasnormaldistributionfunction Itcanbeexpressedas where x probabilityintegralfunction Thisintegralisdifficulttocalculate Usuallytable lookupmethodisusedinsteadofcalculation 45 NormaldistributionexpressedbyerrorfunctionDefinitionoferrorfunction DefinitionofComplementaryerrorfunction Expressionofnormaldistribution 46 频率近似为fc 2 8Narrowbandrandomprocess 2 8 1BasicconceptofnarrowbandrandomprocessWhatdoesitmeannarrowband Assumethebandwidthofarandomprocessis f thecentralfrequencyisfc If f fc thentherandomprocessiscalledanarrowbandrandomprocess WaveformandexpressionofnarrowbandrandomprocessWaveformandspectrum 47 Expressionwhere aX t randomenvelopeofnarrowbandrandomprocess X t randomphaseofnarrowbandrandomprocess 0 angularfrequencyofsinusoidalwaveTheaboveequationcanberewrittenas where inphasecomponentofX t orthogonalcomponentofX t 48 2 8 2CharacteristicsofnarrowbandrandomprocessStatisticalcharacteristicsofXc t andXs t IfX t isastationarynarrowbandGaussianprocesswithzeromean thenXc t andXs t arealsoGaussianprocesses Xc t andXs t haveidenticalvariance andthevarianceisequaltothevarianceofX t XcandXsatthesameinstantareuncorrelatedandstatisticallyindependent StatisticalcharacteristicsofaX t and X t ProbabilitydensityofaX t Probabilitydensityof X t 49 2 9SinusoidalwaveplusnarrowbandGaussianprocess Expressionofsinusoidalwaveplusnoise where A deterministicamplitudeofsinusoidalwave 0 angularfrequencyofsinusoidalwave randomphaseofsinusoidalwaven t narrowbandGaussiannoiseProbabilitydensityoftheenvelopeofr t where 2 varianceofn t I0 zero ordermodifiedBesselfunctionpr x iscalledgeneralizedRayleighdistribution orRiciandistribution WhenA 0 pr x becomesRayleighprobabilitydensity 50 Conditionalprobabilitydensityofthephaseofr t where phaseofr t includingthephase ofsinusoidalwaveandthephaseofnoisepr conditionalprobabilitydensityofthephaseofr t undertheconditionofgiver Probabilitydensityofthephaseofr t When 0 where 51 CurvesofRiciandistributionWhenA 0 Envelope RayleighdistributionPhase UniformdistributionWhenA isverylarge Envelope normaldistributionPhase impulsefunction 52 2 10Signaltransferthroughlinearsystems 2 10 1BasicconceptoflinearsystemsCharacteristicsofthelinearsystemsdiscussedhereHaveapairofinputandapairofoutputPassiveMemorylessTime invariantCausalityLinear satisfyingsuperpositionprincipleIfwheninputisxi t outputisyi t thenwheninputistheoutputiswhere a1anda2arebotharbitraryconstants 53 Sketchoflinearsystem 54 2 10 2DeterministicsignaltransferthroughlinearsystemsTimedomainanalysismethodLeth t impulseresponseofthesystemx t inputsignalwaveformy t outputsignalwaveformthenwehave 55 FrequencydomainanalysismethodAssumeinputisaenergysignal letx t inputenergysignalH f Fouriertransformofh t X f Fouriertransformofx t y t outputsignalthenthefrequencyspectraldensityY f oftheoutputsignaly t ofthesystemis y t canbefoundfromtheinverseFouriertransformofY f 56 Assume theinputx t isaperiodicpowersignal thenwhere theoutputis Iftheinputx t isanonperiodicalpowersignal thenitwillbeprocessedasarandomsignal 57 Example2 10 ThereisaRClow passfilterasshowninFig 2 10 4 Finditsimpulseresponseandtheexpressionofitsoutputsignalwhentheinputisexponentiallyattenuated Solution Assumex t inputenergysignaly t outputenergysignalX f frequencyspectraldensityofx t Y f frequencyspectraldensityofy t thenthetransferfunctionofthecircuitis 58 Theimpulseresponseh t ofthefilter Therelationshipbetweentheoutputandtheinputofthefilter Assumetheinputx t equals thentheoutputofthefilteris 59 ConditionsofdistortionlesstransmissionThereisadistortionlesslineartransmissions