2.2随机过程自相关函数与功率谱Theself-correlationfunctions&powerspectraofRPs信号频谱和傅立叶变换Thespectra&Fouriertransformsofsignals1、基本概念Basicconcepts（1）实信号：可用时间实函数表示信号Realsignal:Thesignalsexpressedwithrealfunctionoftimeare….特点：含有有限能量或有限功率Features:Theenergyorpowerofarealsignalis`finite.（2）能量信号：能量有限信号Energysignal:Thesignalswithfiniteenergies（3）时间函数信号分解Thedecompositionoftimefunctionsignals<1>一个时间函数信号可表示成若干个基本信号总和或积分Atimefunctionsignalcanbeexpressedwiththesumorintegralofacertainnumberofbasicsignals<2>惯用基本信号：复正弦信号、函数、sinc函数等Thebasicsignalsfrequentlyused:complexsinesignal,function,sincfunction(samplefunction)etc.（4）时间函数信号频谱密度---傅立叶变换Thespectrumdensityoftimefunctionsignals---FourierTransform当用复正弦信号作为基本信号时，以时间函数表示信号可写成(反傅立叶变换形式)(1.2.21)其中(1.2.22)称为频谱密度或傅立叶变换;称为傅立叶反变换,并将这种关系记为(1.2.23)Whencomplexsinesignalsareusedasbasicsignals,atimefunctionsignalcanbewrittenwiththeformofInverseFourierTransformasWhereiscalledthespectrumdensity,ortheFourierTransformofthe,andtheistheInverseFourierTransformofthe.Thisrelationisdenotedas.ThetwofunctionsarecalledaFTpair.2、傅立叶变换主要特征TheimportantpropertiesofFT（1）线性性质Linearity若函数、…所对应傅立叶变换分别是、…，，则以下变换对成立：(1.2.24)式中为有限正整数,为常系数。Thefollowingequalitywillholdif,,…arethecorrespondingFouriertransformsof,,…,respectively:Whereisanintegerandsareconstantcoefficients.（2）尺度性质Scaletransformation若,则对实常数有(1.2.25)If,thenforarealconstant,thefollowingequalityholds:（2）单位阶跃函数Unitjumpfunction<1>Definition:(1.2.40)<2>FT:(1.2.41)（3）指数函数Exponentialfunction（）Prove:AccordingtothefrequencyshiftingfeatureandWehaveIfdefinethenwehave(1.2.49)<2>巴塞瓦公式ParsevalFormulaWhen,formula(1.2.48-2)becomes(1.2.50)ThisiscalledParsevalFormula,whichisthemeasurementoftheextentofcorrelationoftwosignalsinfrequencydomain.<3>自相关函数傅立叶变换及其能谱密度函数TheFTandtheEnergySpectrumDensityFunctionsofself-correlationfunctionsSubstitutingthesubscriptybyxinForm.（）,wehave(1.2.51)Thereforewecandenote(1.2.52)wheretheiscalledtheEnergySpectrumDensity(能谱密度)of.物理含义:能量信号自相关