随机过程的自相关函数与功率谱.ppt

2 2随机过程的自相关函数与功率谱Theself correlationfunctions powerspectraofRPs信号的频谱和傅立叶变换Thespectra Fouriertransformsofsignals1 基本概念Basicconcepts 1 实信号 可用时间的实函数表示的信号Realsignal Thesignalsexpressedwithrealfunctionoftimeare 特点 具有有限的能量或有限的功率Features Theenergyorpowerofarealsignalis finite 2 能量信号 能量有限的信号Energysignal Thesignalswithfiniteenergies 下一页 3 时间函数信号的分解Thedecompositionoftimefunctionsignals一个时间函数信号可表示成若干个基本信号的总和或积分Atimefunctionsignalcanbeexpressedwiththesumorintegralofacertainnumberofbasicsignals常用基本信号 复正弦信号 函数 sinc函数等Thebasicsignalsfrequentlyused complexsinesignal function sincfunction samplefunction etc 4 时间函数信号的频谱密度 傅立叶变换Thespectrumdensityoftimefunctionsignals FourierTransform 当用复正弦信号作为基本信号时 以时间函数表示的信号可写成 反傅立叶变换的形式 1 2 21 其中 1 2 22 称为的频谱密度或的傅立叶变换 称为的傅立叶反变换 并将这种关系记为 1 2 23 Whencomplexsinesignalsareusedasbasicsignals atimefunctionsignalcanbewrittenwiththeformofInverseFourierTransformas Whereiscalledthespectrumdensity ortheFourierTransformofthe andtheistheInverseFourierTransformofthe Thisrelationisdenotedas ThetwofunctionsarecalledaFTpair 2 傅立叶变换的重要特性TheimportantpropertiesofFT 1 线性性质Linearity若函数 所对应的傅立叶变换分别是 则下列变换对成立 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 3 时延性质TimeDelay若 则有 1 2 26 If thenthefollowingequalityholds 4 频移性质FrequencyShift若 则有 1 2 27 If thenthefollowingequalityholds 5 时域微分与积分Differentialandintegralintimedomain若 则下列各式成立If thenfollowingequalitieshold 1 2 28 1 2 29 1 2 30 若在区间上积分为零 即信号无直流分量 则上式化简为 1 2 31 6 时间倒置TimeReverse若 则有 1 2 32 7 对偶性Duality若 则有 1 2 33 8 时域卷积Timedomainconvolution若 则有 1 2 34 9 频域卷积Frequencydomainconvolution若 则有 1 2 35 1 或记为 1 2 35 2 10 复共轭特性Complexconjugation若 则有 1 2 36 1 2 37 3 典型函数的傅立叶变换TheFToftypicalfunctions 1 单位脉冲函数 函数 Unitpulsefunction function Definition 1 2 38 moreoverFeature FT 1 2 39 ordenotedasInverseFT 2 单位阶跃函数UnitjumpfunctionDefinition 1 2 40 FT 1 2 41 3 指数函数Exponentialfunction 1 2 42 Prove AccordingtothefrequencyshiftingfeatureandWehave 4 正弦与余弦函数SineandCosinefunctions 1 2 43 Similarly 1 2 44 5 振幅为A宽度为T 中心位于原点的矩形脉冲函数TherectangularpulsefunctionwiththeamplitudewidthT centeratorigini e 1 2 45 相关函数和功率TheCorrelationFunctions Power1 相关函数的普遍定义 应以遍历过程为条件 Thegeneraldefinitionofcorrelationfunction condition ergodicprocess 自相关函数Theself correlationfunction 1 2 46 互相关函数Themutualcorrelationfunction 1 2 47 物理含义 两个信号之间的交迭程度 相关程度 两信号完全不交迭时积分为零 完全交迭时积分值最大 部分交迭时积分值介于零与最大值之间 Physicalmeaning Describingtheextentoftheoverlapping correlated betweentwosignals theintegralwillbezerowhenthetwosignalsarenotoverlappedthoroughly maximumwhentheyareoverlappedthoroughly betweenzeroandmaximumwhentheyareoverlappedpartly 2 相关函数的傅立叶变换TheFTofcorrelationfunctions互相关函数的傅立叶变换TheFTofself correlationfunctions 1 2 48 Deriving Ifdefinethenwehave 1 2 49 巴塞瓦公式ParsevalFormulaWhen formula 1 2 48 2 becomes 1 2 50 ThisiscalledParsevalFormula whichisthemeasurementoftheextentofcorrelationoftwosignalsinfrequencydomain 自相关函数的傅立叶变换及其能谱密度函数TheFTandtheEnergySpectrumDensityFunctionsofself correlationfunctions SubstitutingthesubscriptybyxinForm 1 2 49 wehave 1 2 51 Thereforewecandenote 1 2 52 wheretheiscalledtheEnergySpectrumDensity 能谱密度 of 物理含义 能量信号的自相关函数与能量谱密度函数构成傅立叶变换对 Physicalmeaning Theself correlationfunctionandtheenergyspectrumdensityfunctioncomposeaFouriertransformpair 3 能量型复信号和实信号的能量公式TheEnergyFormulaofenergy typedcomplexandrealsignals复信号的能量公式 TheEnergyFormulaofcomplexsignalsWhen accordingtothedefinitionofself correlationfunctionandForm 1 2 51 wehave 1 2 53 Formula 1 2 53 iscalledtheEnergyFormulaofcomplexsignals 复信号的能量公式 实信号的能量公式 TheEnergyFormulaofrealsignalsWhen accordingtothedefinitionofself correlationfunctionandForm 1 2 51 wehave 1 2 54 Formula 1 2 54 iscalledtheEnergyFormulaofrealsignals 在其它书 数理统计 中 能量公式 1 2 53 1 2 54 被称为巴塞瓦公式Physicalmeaning Theleftsideoftheequalitysignistheintegralofsignalpowerintimedomain i e theenergyofthesignal therightsideistheintegralofthesquareofthemodulusofthefrequencyspectrumofthesignalinfreq domain whichisalsotheenergy ThereforethesquareofthemodulusofthefrequencyspectrumiscalledastheEnergySpectrumDensityofthesignal 4 功率型信号的相关函数与功率谱Thecorrelationfunctions powerspectraofpower typedsignals 1 两种类型的信号Twokindsofsignals能量型信号 在整个信号存在的时间内 信号能量为有限值 但平均功率趋于零 Energy typedsignals Theenergyofasignalisfiniteandtheaveragepowerapproachestozerointheexistingtimeofit 功率型信号 在区间内信号功率有限而能量无限的信号 Power typedsignals Thepowerofasignalisfiniteandtheenergyisinfiniteintheexistingtimeofit 2 功率型信号的自相关函数Theself correlationfunctionofpower typedsignals 1 2 55 Physicalmeaning Theaveragepowerofthesignal 信号的平均功率 3 功率型信号的傅立叶变换 功率谱密度TheFTofpower typedsignals PowerSpectrumDensity 1 2 56 1 i e 1 2 56 2 WheretheiscalledthePowerSpectrumDensityofsignal 上式表明 功率型信号的自相关函数与其功率谱密度函数构成傅立叶变换对 Meaning Theself correlationfunctionandpowerspectrumdensityFunctionofapower typedsignalcomposeaFTpair 4 功率型信号的平均功率Theaveragepowerofapower typedsignalletinForm 1 2 56 1 wehave 1 2 57 注意到 求过程中由于是能量无限的 故不可能求出确切的频谱 为此须先定义一个持续时间有限的截短函数 1 2 58 使得成为能量有限函数 可有确切的频谱 即有令则有 1 2 59 类比于能量型信号的关系 对功率型信号有 1 2 60 故有 1 2 61 Itshouldbenoticedthatintheprocedureofsolving Itisimpossibletoobtainanexactfrequencyspectrumfunction becausethattheenergyofisinfinite Therefore atruncatedfunctionlastingafinitepieceoftimeshouldbedefined inadvance asTheisatimefinitefunction whichhasanexactfrequencyspectrumfunction andtherelation Let WehaveOntheanalogyoftherelationbetweentheself correlationandenergyspectrumdensityoftheenergy typedsignals wehaveforpower typedsignals Letandsolvethelimitofaboveformula thenwehavetheconclusion 5 随机过程样本函数的功率谱ThepowerspectrumofthesamplefunctionsofRPs随机过程样本函数是功率型函数 但考虑到其频谱的随机性 在求其功率谱时还须对作统计平均 故随机过程 的功率谱密度公式成为其中 1 2 62 式中表示求统计平均 若对谱大小不感兴趣可将忽略 ThesamplefunctionofaRPispower typed Consideringtherandompropertyofitstruncatedfrequencyspectrum thestatisticalaveragingofshouldbecarriedwhensolvingitspowerspectrum ThereforetheformulaofthepowerspectrumdensityofaRPbecomeswhereThe expressesstatisticalaverageandthecanbeneglectedwhenthemagnitudeofisnotinteresting 6 小结Summary Importantrelations 频谱密度TheFrequencySpectrumDensityFunction 能量谱密度TheEnergySpectrumDensityFunction 能量公式TheEnergyFormula ParsevalFormulainotherbooks 功率谱密度ThePowerSpectrumDensityFunction 平均功率TheAveragePowerofaRP