您好,欢迎访问三七文档
当前位置:首页 > 电子/通信 > 综合/其它 > Principle-of-Communication-System-5.2
PrinciplesofCommunicationSystemBasicConceptofNarrowBandRandomProcess:NarrowBandRandomProcessWhatdoesitmeannarrowband?∆𝒇≪𝒇𝒄WaveformandexpressionofnarrowbandrandomprocessWaveformandspectrum:频率近似为fcBasicConceptofNarrowBandRandomProcess:NarrowBandRandomProcessExpression:𝑿𝒕=𝒂𝑿𝒕𝐜𝐨𝐬𝝎𝟎𝒕+𝝋𝑿(𝒕),𝒂𝑿≥𝟎Where,𝒂𝑿(𝒕)—randomenvelopeofNBRP𝑿(𝒕)𝝋𝑿(𝒕)—randomphaseofNBRP𝑿(𝒕)𝝎𝟎—angularfrequencyofsinusoidalwaveTheaboveequationcanberewrittenas:𝑿𝒕=𝑿𝒄𝒕𝐜𝐨𝐬𝝎𝟎𝒕−𝑿𝒔𝒕𝐬𝐢𝐧𝝎𝟎𝒕Where,𝑿𝒄𝒕=𝒂𝑿𝒕𝐜𝐨𝐬𝝋𝑿(𝒕)—inphasecomponentof𝑿(𝒕)𝑿𝒔𝒕=𝒂𝑿𝒕𝐬𝐢𝐧𝝋𝑿(𝒕)—orthogonalcomponentof𝑿(𝒕)CharacteristicsofNarrowBandRandomProcess:NarrowBandRandomProcessIf𝑿(𝒕)isastationarynarrowbandGaussianprocesswithzeromean,then𝑿𝒄𝒕and𝑿𝒔𝒕arealsoGaussianprocesses.Statisticalcharacteristicsof𝑿𝒄𝒕and𝑿𝒔𝒕𝑿𝒄𝒕and𝑿𝒔𝒕haveidenticalvariance,andthevarianceisequaltothevarianceof𝑿(𝒕).𝑿𝒄and𝑿𝒔atthesameinstantareuncorrelatedandstatisticallyindependent.CharacteristicsofNarrowBandRandomProcess:NarrowBandRandomProcessProbabilitydensityof𝒂𝑿𝒕:Statisticalcharacteristicsof𝒂𝑿𝒕and𝝋𝑿𝒕𝑷𝒂𝑿=𝒂𝑿𝝈𝑿𝟐𝐞𝐱𝐩−𝒂𝑿𝟐𝟐𝝈𝑿𝟐𝒂𝑿≥𝟎Probabilitydensityof𝝋𝑿𝒕:𝑷𝝋𝑿=𝟏𝟐𝝅𝟎≤𝝋𝑿≤𝟐𝝅Sinusoidalwave+NarrowBandGaussianProcessExpressionofsinusoidalwaveplusnoise:𝒓𝒕=𝑨𝐜𝐨𝐬𝝎𝟎𝒕+𝜽+𝒏(𝒕)𝒑𝒓𝒙=𝒙𝝈𝟐𝑰𝟎𝑨𝒙𝝈𝟐𝒆𝒙𝒑−𝟏𝟐𝝈𝟐𝒙𝟐+𝑨𝟐,𝒙≥𝟎Probabilitydensityoftheenvelopeof𝒓(𝒕):Where,𝝈𝟐—varianceof𝒏(𝒕)𝑰𝟎(⋅)—zero-ordermodifiedBesselfunction𝒑𝒓(𝒙)—generalizedRayleighdistribution,orRicedistribution.When𝑨=𝟎,𝒑𝒓(𝒙)becomesRayleighprobabilitydensity.Sinusoidalwave+NarrowBandGaussianProcessConditionalprobabilitydensityofthephaseof𝒓(𝒕):𝒑𝒓𝝋/𝜽=𝐞𝐱𝐩(−𝑨𝟐𝟐𝝈𝟐)𝟐𝝅+𝑨𝐜𝐨𝐬(𝜽−𝝋)𝟐(𝟐𝝅)𝟏𝟐𝝈𝐞𝐱𝐩−𝑨𝟐𝟐𝝈𝟐𝐬𝐢𝐧𝟐(𝜽−𝝋)𝟏+𝒆𝒓𝒇𝑨𝐜𝐨𝐬(𝜽−𝝋)𝟐𝟏𝟐𝝈Probabilitydensityofthephaseof𝒓(𝒕):𝒑𝒓𝝋=𝒑𝒓𝝋𝜽𝒑𝒓𝜽𝒅𝜽𝟐𝝅𝟎When𝜽=𝟎,then𝒑𝒓𝝋𝟎=𝟏𝟐𝝅𝐞𝐱𝐩−𝑨𝟐𝟐𝝈𝟐𝟏+𝑮𝝅𝟏+𝒆𝒓𝒇(𝑮)𝐞𝐱𝐩𝑮𝟐𝟎≤𝝋≤𝟐𝝅Where,𝑮=𝑨𝐜𝐨𝐬𝝋𝟐𝝈𝒆𝒓𝒇(𝑮)=𝟐𝝅𝒆−𝒕𝟐𝒅𝒕𝑮𝟎rRayleighdistributionProbabilitydensityEnveloper(a)ProbabilitydensityoftheenvelopewithRicedistributionSinusoidalwave+NarrowBandGaussianProcessCurvesofRicedistribution:When𝑨𝝈=𝟎,thenenvelope→Rayleighdistributionphase→UniformdistributionWhen𝑨𝝈≫𝟏,thenenvelope→Normaldistributionphase→ImpulsefunctionSinusoidalwave+NarrowBandGaussianProcessCurvesofRicedistribution:(b)ProbabilitydensityofthephasewithRiciandistributionUniformphasePhaseProbabilitydensityBasicconceptofLinearSystems:SignaltransferthroughLinearSystemsHaveapairofinputandapairofoutputIfwheninputis𝒙𝒊(𝒕),outputis𝒚𝒊(𝒕),thenwheninputis:Characteristicsofthelinearsystems:PassiveMemorylessTime-invariantCausalityLinear:satisfyingsuperpositionprinciple𝒙𝒕=𝒂𝟏𝒙𝟏𝒕+𝒂𝟐𝒙𝟐𝒕theoutputis𝒚𝒕=𝒂𝟏𝒚𝟏𝒕+𝒂𝟐𝒚𝟐𝒕BasicconceptofLinearSystems:SignaltransferthroughLinearSystemsSketchofLinearSystems:LinearSystemInputOutputx(t)y(t)X(f)Y(f)h(t)H(f)t(t)h(t)t00DeterministicsignaltransferthroughLinearSystems:SignaltransferthroughLinearSystemsTimedomainanalysismethodForphysicallyrealizablesystems:𝒉(𝒕)𝒅𝒕∞∞−∞𝒉𝒕=𝟎,𝒕𝟎Thenwehave:𝒚𝒕=𝒙(𝒕)∗𝒉(𝒕)=𝒙𝝉𝒉𝒕−𝝉𝒅𝝉∞−∞=𝒙𝒕−𝝉𝒉𝝉𝒅𝝉∞−∞DeterministicsignaltransferthroughLinearSystems:SignaltransferthroughLinearSystemsFrequencydomainanalysismethodAssumeinputisaenergysignal,let:𝒙𝒕↔𝑿(𝒇)Thenwehave:𝒀(𝒇)=𝑿𝒇∙𝑯(𝒇)𝒚𝒕↔𝒀(𝒇)𝒉𝒕↔𝑯(𝒇)𝒚(𝒕)canbefoundfromtheinverseFouriertransformof𝒀(𝒇)𝒚𝒕=𝒀(𝒇)𝒆𝒋𝝎𝒕𝒅𝒇∞−∞DeterministicsignaltransferthroughLinearSystems:SignaltransferthroughLinearSystemsFrequencydomainanalysismethodAssumeinputisaperiodicpowersignal,then:𝒙𝒕=𝑪(𝒋𝒏𝝎𝟎)𝒆𝒋𝒏𝝎𝟎𝒕∞𝒏=−∞theoutputis:Iftheinput𝒙(𝒕)isanonperiodicalpowersignal,thenitwillbeprocessedasarandomsignal.𝒚𝒕=𝑪𝒋𝒏𝝎𝟎𝑯(𝒏𝝎𝟎)𝒆𝒋𝒏𝝎𝟎𝒕∞𝒏=−∞Example2.10ThereisaRClow-passfilterasshowninFig.2.10.4.Finditsimpulseresponseandtheexpressionofitsoutputsignalwhentheinputisexponentiallyattenuated.Solution:Assume𝒙𝒕—inputenergysignal𝒚𝒕—outputenergysignal𝑿(𝒇)—frequencyspectraldensityof𝒙𝒕𝒀(𝒇)—frequencyspectraldensityof𝒚𝒕thetransferfunctionofthecircuitis:𝑯𝒇=𝟏𝒋𝝎𝑪𝑹+(𝟏𝒋𝝎𝑪)=𝟏𝟏+𝒋𝝎𝑹𝑪Fig2.10.4RCfilterRCx(t)y(t)Theimpulseresponseh(t)ofthefilter:𝒉𝒕=𝑯(𝒇)𝒆𝒋𝝎𝒕𝒅𝒇∞−∞=𝟏𝟏+𝒋𝝎𝑹𝑪𝒆𝒋𝝎𝒕𝒅𝒇∞−∞=𝟏𝑹𝑪𝒆−𝒕𝑹𝑪Therelationshipbetweentheoutputandtheinputofthefilter:𝒚𝒕=𝒙𝒕∗𝒉𝒕=𝒙𝝉𝒉𝒕−𝝉𝒅𝝉∞−∞=𝟏𝑹𝑪𝒙(𝝉)𝒆−(𝒕−𝝉)𝑹𝑪𝒅𝝉∞−∞Assumetheinput𝒙(𝒕)equals:𝒙𝒕=𝒆−𝒂𝒕,𝒕≥𝟎𝟎,𝒕𝟎thentheoutputofthefilteris:𝒚𝒕=𝟏𝑹𝑪𝒆−𝒂𝝉𝒆−(𝒕−𝝉)𝑹𝑪𝒅𝝉𝒕𝟎=𝒆−𝒕𝑹𝑪𝑹𝑪∙𝒆𝝉𝟏𝑹𝑪−𝒂𝟏𝑹𝑪−𝒂𝒕𝟎=𝒆−𝒂𝒕−𝒆−𝒕𝑹𝑪𝟏−𝒂𝑹𝑪DeterministicsignaltransferthroughLinearSystems:SignaltransferthroughLinearSystemsConditionsofdistortionlesstransmissionAssumeinputisanenergysignal𝒙(𝒕),then:𝒚𝒕=𝒌𝒙(𝒕−𝒕𝒅)Findthetransferfunctionofthesystem:Where𝜽=𝟐𝝅𝒇𝒕𝒅∴𝑯𝒇=𝒌𝒆−𝒋𝝎𝒕𝒅=𝒌𝒆−𝒋𝜽𝒀𝒇=𝑿𝒇∙𝑯𝒇=𝒌𝑿(𝒇)𝒆−𝒋𝝎𝒕𝒅|H(f)|k0ff0AmplitudecharacteristicisindependentoffrequencyPhasecharacteristicisastraightlinevi
本文标题:Principle-of-Communication-System-5.2
链接地址:https://www.777doc.com/doc-5427620 .html