第一讲随机过程

随机信号分析（国际班）——第一讲1LectureOneRandomProcess13.ProbabiltydistributionofnarrowbandprocessenvolopingsquareTips:Wehavetalkedabouttheprobabiltydistributionofenvolopingandphaseofnarrow-bandprocessinbothcasesofpurenoiseandsinusoidalsignalplusnoiseandunderstoodthedistributionformofthesignal,whichcandeterminethedetectionthresholdspecificallyandfindsignalfromthenoisebackground.Detectionmethodcanbeusedforphaseorenvelope,buttheenvelopedetectionwasusedcommonlybecauseitisrelativelyeasytoimplement.Initialphaseofincoherentsignalswhichweusuallyusedcan’tusephasedetectionbecauseofitsrandomchange.Therefore,theenvelopedetectionisdividedintolineardetectionandsquare-lawdetection.Theformerhasmorecomplextheoreticalanalysis,butpracticesmoreanditisbasedontheprobabilitydistributionoftheenvelopeofnarrowbandprocess.Incontrasttotheformer,thelatterusesmulti-square-lawdetectioninthetheoreticalanalysisanditisbasedontheprobabilitydistributionofthesquareoftheenvelopeofnarrowbandprocess.Infact,thesetwodetectorshavelittledifferenceinperformance.Whentheinformationprocessingneedstoknowtheprobabilitydistributiondensityafterdetection,theresultsdiscussedaboveisverypractical.1.(Purenoise)probabilitydistributionoftheenvelopesquareofnarrowbandnormalnoisenarrowbandnormalnoisecanberepresentedas:tSintAtCostAttCostAtnSC000narrowbandnormalnoiseenvelope:随机信号分析（国际班）——第一讲2tAtAtASC22uA0TheprobabilitydensityofnarrowbandnormalnoiseenvelopetAis2222)(AeAAp——RayleighdistributionIfenvelopedetectorhashalf-wavesquare-lawdetectioncharacteristics,thatis,theoutputvoltagetAtu20,AuAccordingtothetransformofrandomvariablefunction,theenvelopesquareuoftheprobabilitydensityis222221AeAAApdudAup22221ue0u——exponentialdistributionMakethenormalizedrandomvariableis2uv,theprobabilitydensityis221veupdvduvp0v——exponentialdistributionstandardexpression2.Narrowbandnormalnoise+probabilitydistributionofenvelopesquareinthesinusoidalsignalsynthesisprocess随机信号分析（国际班）——第一讲3synthesisprocessis：ttCostRtntstY0TheprobabilitydensityoftheenvelopetRinsynthesisprocess:2022222expARIARRRp0R——ObeythegeneralizedRayleighdistributionIfenvelopedetectorhashalf-wavesquare-lawdetectioncharacteristics,thatis,theoutputvoltagetRtu20R,uAccordingtotheoftransformrandomvariablefunction,theprobabilitydensityoftheenvelopesquareuisRpdudRup202222exp21AuIAu0u令2uv，AvIvupdvduvpA02exp21220v3.2distributionandnon-Center2distribution(Greekalphabet[kai])（1）AccumulationofthevideosignalimprovesthedetectionperformanceTips:Whenweuseenvelopedetectionmethodtodetecttheperiodicsignalnoiseinthebackground,weusuallychoosevideoaccumulationin随机信号分析（国际班）——第一讲4ordertoimprovedetectionperformance.Tips:weneedtodeterminewhethertheoutputofthereceiverhastheinformationwhichisdesiredtocheckinsignaldetectiontheory.However,signalisalwaysaccompaniedbynoiseandtheyaremixedtogethertoformarandomprocess.Inordertoimprovethedetectionperformance,weoftenaddupsamplevalueswhichareoutputtedbysquarelawdetector(accumulation)andthencomparewiththethresholdlevelbeforejudgingwhetherthesignalisappearedornot.Becausethereismorethanonesignalthereceiverreceivinginmanyapplications,forexampleradarisabouttolaunchmanypulseinsequenceatwork,andthereceiverneedtoobservemanyechopulseswhicharereceivedatdifferenttimesbeforejudging.Theseechopulsesusuallycarriethesameinformationorallofthereflectioninformationwhichcontainingobject,oronlyhavenoise.Thatistosay,everypulsesignalcontainstheinformationwhethertargetswouldappearandeachpulsesignalnoiseisgenerallystatisticallyindependenttoeachother.Wecancombinethesesignalsbeforejudgingtheseyesornosignalsandthedetectionperformancecanbesignificantlyimproved.Tips:Addupthesignal(accumulation)intwosituations:oneistheadditionbeforedetector,anotheristheadditionafterdetector.Wearegoingtodiscussthelatter.Amplitudedetectorhastwokinds:onekindislineardetector,theotherissquarelawdetector.ThedistributionoflineardetectoroutputprocessisRayleighdistribution,theunifiedexactexpressionscannotbegetbythesumofmultiplegeneralizedRayleighdistribution,sowearenotresearchitTheoutputofsquarelawdetectorisdifferent.whenitsinputcompletelyhasRayleighenvelopestatisticalproperties(narrowbandgaussiannoise),thesumof(non-coherentaccumulationofnpulses)itsoutputstatisticsindependentsamplingvalues(condition:samplingtimeislongenough,so随机信号分析（国际班）——第一讲5thatthesamplingvaluesareindependent)willproducetheprobabilitydensityfunctiondistributioncontainingndegreesoffreedom2.窄带滤波器正弦信号加宽带噪声平方律检波器21独立取样n次加法器n1v)()()(222tAtAtRsc)](cos[)()(0tttRtY(归一化)tSintnASintCostnACostntstYSC00tACtASTips:Inthediagram,signalplusnoiseisthroughthenarrowbandsystemandtYbecomessignalwithnarrowbandnormalprocess.ItisthroughthesquarelawdetectoranddoesenvelopedetectionandgetssquareenvelopeprocesstAtAtRSC222.。Nomatterwhetherthesignalexists,tACandtASarenormaldistribution.Thetwoprocessoflowfrequency(video)donor