1CH1：RandomProcessesIntroductionMathematicalDefinitionofaRandomProcessStationaryProcessesMean,Correlation,andCovarianceFunctionsErgodicProcessesTransmissionofaRandomProcessThroughaLinearTime-InvariantFilterPowerSpectralDensityGaussianProcessNoiseNarrowbandNoiseRepresentationofNarrowbandNoiseinTermsofIn-phaseandQuadratureComponentsRepresentationofNarrowbandNoiseinTermsofEnvelopeandPhaseComponentsSineWavePlusNarrowbandNoiseComputerExperiments:Flat-FadingChannelSummaryandDiscussion21.1IntroductionTwomathematicalmodelsDeterministicStochastic(orrandom)Receivedsignalinacommunicationsystemusuallyconsistsof:Information-bearingsignalRandominterferenceChannelnoise

DescribingthesignalusingstatisticalparametersAveragepower,powerspectraldensity,…3Random(stochastic)processPropertiesFunctionoftimeRandomDefinitionEnsembleoftimefunctionsAprobabilityrule1.2MathematicalDefinitionofaRandomProcess41.2MathematicalDefinitionofaRandomProcess(Cont’d)Figure1.1Anensembleofsamplefunctions.Someconcepts:SamplespaceSRandomprocessX(t,S)=X(t)SamplepointsjRealization(samplefunction)

xj(t)=X(t,sj)Randomvariable51.3StationaryProcessThejointdistributionfunction:Strictlystationary：Foralltimeshifts

,allk,andallpossiblechoicesofobservationtimest1,…,tk,equation(1)isalwaystrue.Twospecialcases(wide-sensestationary):6Example1.1Three

spatialwindowslocatedattimest1,t2,andt3,theprobabilityofthejointevent:

Intermsofthejointdistributionfunction,thisprobabilityequals:1.3StationaryProcessFigure1.2Illustratingtheprobabilityofajointevent.71.3StationaryProcessFigure1.3IllustratingtheconceptofstationaryinExample1.1.81.4Mean,Correlation,andCovarianceFunctionsMean:Autocorrelationfunction:Autocovariancefunction:(Stationary)Cross-correlationfunction:91.4Mean,Correlation,andCovarianceFunctions(Cont’d)Themeanandautocorrelationfunctionprovideapartialdescriptionofarandomprocess.Wide-sensestationaryMeanisaconstantandautocorrelationfunctiondependsonlyontimedifference.OftenusedinpracticeNotnecessarystrictlystationary,andviseverse.10PropertiesoftheAutocorrelationFunctionProperties:DefiningautocorrelationfunctionofastationaryprocessX(t)as:11PropertiesoftheAutocorrelationFunction(Cont’d)Figure1.4Illustratingtheautocorrelationfunctionsofslowlyandrapidlyfluctuatingrandomprocesses.12Example1.2SinusoidalWavewithRandomPhaseAandfcareconstants,and13Example1.2(Cont’d)TheautocorrelationfunctionofX(t)is:14Example1.2(Cont’d)

Figure1.5Autocorrelationfunctionofasinewavewithrandomphase.15Example1.3RandomBinaryWave

Figure1.6Samplefunctionofrandombinarywave.16Example1.3(Cont’d)Figure1.7Autocorrelationfunctionofrandombinarywave.17Cross-CorrelationFunctionsTworandomprocessesX(t)andY(t)withautocorrelationfunctionsRX(t,u)andRY(t,u),thetwocross-correlationfunctionsofX(t)andY(t)aredefinedby:Thecorrelationmatrix:Asymmetryrelationship:18Example1.4Quadrature-ModulatedProcesses

19Example1.4(Cont’d)

201.5ErgodicProcessesUsingtimeaveragestoapproximateensembleaverages.Consideringasamplefunctionx(t)ofastationaryprocessX(t)inanobservationwindow–TtT:(TheDCvalue)TimeaverageX(T)

representsanunbiasedestimateoftheensemble-averagedmeanX.211.5ErgodicProcesses(Cont’d)AprocessX(t)isergodicinthemeaniftwoconditionsaresatisfied:AprocessX(t)isergodicintheautocorrelationfunctioniftwoconditionsaresatisfied:Forarandomprocesstobeergodic,ithastobestationary;butastationaryrandomprocessisnotnecessarilyergodic.221.6TransmissionofaRandomProcessThroughaLinearTime-InvariantFilterFigure1.8transmissionofarandomprocessthroughalineartime-invariantfilter.231.6TransmissionThroughaLinearTime-InvariantFilter(Cont’d)241.7PowerSpectralDensity(PSD)25DefinitionofPSDThepowerspectraldensity(orpowerspectrum)istheFouriertransformoftheautocorrelationfunction.Asaresult,IfThenandfissmall,26PropertiesofPSDThePSDandtheautocorrelationfunctionformaFourier-transformpair.TheEinstein-Wiener-KhintchineRelations27PropertiesofPSD(Cont’d)isaprobabilitydensityfunction.28PSDExample1Sinusoidalwavewithrandomphase29PSDExample1(Cont’d)Figure1.10Powerspectraldensityofsinewavewithrandomphase.30PSDExample2Randombinarywave31PSDExample2(Cont’d)Figure1.11Powerspectraldensityofrandombinarywave.32PSDExample3Mixingofarandomprocesswithasinusoidalprocess33PSD’sofInput/OutputProcesses34PSDandtheMagnitudeSpectrumWeareconsideringanergodicstationaryprocess.Fouriertransformablerequiresabsolutelyintegrable,thatiswhichcannotbesatisfiedbyastationaryfunction.Soweuseatruncatedsegmentofx(t),whoseFouriertransformis35PSDandtheMagnitudeSpectrum(Cont’d)ThePeriodogram36PSDandtheMagnitudeSpectrum(Cont’d)37Cross-SpectralDensities(CSD)Properties38CSDExample139ConceptsStatisticallyindependentanduncorrelatedStatisticallyindependent:F(X,Y)=F(X)F(Y)Uncorrelated:CXY()=0Independentstatisticsarealwaysuncorrelated,buttheconverseisnotnecessarilytrue.40CSDExample2Figure1.12Apairofseparatelineartime-invariantfilters.411.8GaussianProcessDefinition:SupposeSisthesetoflinearfunctionalsofarandomprocessX(t)withfinitemean-squarevalue,ifeveryelementinSisaGaussian-distributedrandomvariable,thenX(t)isaGaussianprocess.Inshort,X(t)isaGaussianprocessifeverylinearfunctionalofX(t)isaGaussianrandomvariable.EasytoprocessandfitformanyphysicalphenomenaAlinearfunctionalofX(t)pdfofGaussiandistribution:pdfofnormalizedGaussiandistributionYN(0,1):42GaussianDistributionFigure1.13NormalizedGaussiandistribution.43CentralLimitTheoremIndependentlyandidenticallydistributed(i.i.d.)randomvariablesXi,i=1,2,…TheXiarestatisticallyindependentTheXihavethesameprobabilitydistributionYiarenormalizedversionofXi Yi=(Xi-x)/X,i=1,2,…Thecentrallimittheorem:44PropertiesofaGaussianProcessIftheinputprocesstoastablelinearfilterisGaussian,thentheoutputprocessisalsoGaussian.ThesetofrandomvariablesobtainedbysamplingaGaussianrandomprocessatdifferenttimesarejointlyGaussian.(CanbeusedasadefinitionofGaussianprocess)DeterminantofMeanvectorCovariancematrix45PropertiesofaGaussianProcess(Cont’d)IfaGaussianprocessisstationary,thentheprocessisalsostrictlystationary.IfasetofrandomvariablesobtainedbysamplingaGaussianrandomprocessatdifferenttimeareuncorrelated,thentheyarestatisticallyindependent.461.9NoiseExternalorinternaltothesystemShotnoiseArisingduetothediscretenatureofcurrentflowinsomeelectronicdevicesNumberofarriversinapre-definedintervalfollowsPoissondistributionThermalNoiseArisingduetorandommotionofelectronsinaconductorUsuallymodeledusingtheThéveninequivalentcircuitortheNortonequivalentcircuitAvailablenoisepoweriskTfwatts.At20oC,kT-174dBm/Hz47ModelingThermalNoiseFigure1.15Modelsofanoisyresistor.(a)Théveninequivalentcircuit.(b)Nortonequivalentcircuit.48WhiteNoiseAnidealizedformofnoisefornoiseanalysisofcommunicationsystemsFigure1.16Characteristicsofwhitenoise.(a)Powerspectraldensity.(b)Autocorrelationfunction.Boltzmann’sconstantEquivalentnoisetemperature49WhiteNoise(Cont’d)SamplesatdifferenttimesonawhitenoiseareuncorrelatedIfthewhitenoiseisGaussian(calledwhiteGaussiannoise),thesamplesarealsostatisticallyindependent(theultimaterandomness)Aslongasthebandwidthofanoiseprocessattheinputofasystemisappreciablylargerthanthatofthesystemitself,wemaymodelthenoiseprocessaswhitenoise.50Example1.10IdealLow-PassFilteredWhiteNoiseFigure1.17Characteristicsoflow-passfilteredwhitenoise.(a)Powerspectraldensity.(b)Autocorrelationfunction.51Example1.11CorrelationofWhiteNoisewithaSinusoidalWave52RepresentationsofBand-PassSignals

(Appendix2.3,2.4)Aband-passsignalisdefinedas:Hilberttransform53Band-PassSignals(Cont’d)Pre-envelope54NarrowbandSignalsFig.A2.4

Magnitudespectrumof(a)band-passsignal,(b)pre-envelope,(c)complexenvelope.551.10NarrowbandNoiseFig.1.18(a)Powerspectraldensityofnarrowbandnoise.(b)Samplefunctionofnarrowbandnoise.56NarrowbandNoise(Cont’d)Tworepresentations:In-phaseandquadraturecomponentsEnvelopandphaseEachrepresentationtotallydescribesthenoiseprocess.571.11RepresentationofNarrowbandNoiseinTermsof

In-PhaseandQuadratureComponentsThecanonicalrepresentationofnarrowbandnoisen(t)nI(t):thein-phasecomponentnQ(t):thequadraturecomponentTheyarebothlow-passsignals.Theyarefullyrepresentativeofn(t),exceptfc.58PropertiesoftheIn-PhaseandQuadratureComponentsofaNarrowbandNoiseZeormeanIfn(t)isGaussian,thennI(t)andnQ(t)arejointlyGuassianIfn(t)isstationary,thennI(t)andnQ(t)arejointlystationary59Properties(Cont’d)nI(t)andnQ(t)havethesamepowerspectraldensity60Properties(Cont’d)nI(t)andnQ(t)havethesamevarianceasn(t)Thecross-spectraldensityofnI(t)andnQ(t)ispurelyimaginary61Properties(Cont’d)Ifn(t)isGaussiananditspowerspectraldensitySN(f)issymmetricaboutthemid-bandfrequencyfc,thennI(t)andnQ(t)arestatisticallyindependent.62AnalyzerandSynthesizerFig.1.19(a)Extractionofin-phaseandquadraturecomponentsofanarrowbandprocess.(b)Generationofanarrowbandprocessfromitsin-phaseandquadraturecomponents.63Example1.12:IdealBand-PassFilteredWhiteNoise64Example1.12(Cont’d)Fig.1.20Characteristicsofidealband-passfilteredwhitenoise.

(a)Powerspectraldensity,

(b)Autocorrelationfunction,

(c)Powerspectraldensityofin-phaseandquadraturecomponents.651.12RepresentationofNarrowbandNoiseinTermsofEnvelopeandPhaseComponentsTheenvelopeofn(t)Thephaseofn(t)Theenveloper(t)andphase(t)arebothsamplefunctionsoflow-passrandomprocesses.66ProbabilityDistributionsoftheEnvelope

andPhaseComponentsTheprobabilitydistributionsarederivedfromthoseofNI(t)andNQ(t).67ProbabilityDistributions(Cont’d)Fig.1.21Illustratingthecoordinatesystemforrepresentationofnarrowbandnoise:(a)intermsofin-phaseandquadraturecomponents,and(b)intermsofenvelopeandphase.DefineThen68ProbabilityDistributions(Cont’d)Rayleighdistribution69ProbabilityDistributions(Cont’d)Fig.1.22

NormalizedRayleighdistribution.701.13SineWavePlusNarrowbandNoiseAssumi