随机过程的自相关函数与功率谱

2.2 随机过程的自相关函数与功率谱随机过程的自相关函数与功率谱 The self-correlation functions&power spectra of RPs 1 信号的频谱和傅立叶变换 The spectra&Fourier transforms of signals1、基本概念 Basic concepts（1）实信号：可用时间的实函数表示的信号 Real signal:The signals expressed with real function of time are.特点：具有有限的能量或有限的功率 Features:The energy or power of a real signal is finite.（2）能量信号：能量有限的信号 Energy signal:The signals with finite energies下一页 （3）时间函数信号的分解 The decomposition of time function signals 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 A time function signal can be expressed with the sum or integral of a certain number of basic signals 常用基本信号：复正弦信号、函数、sinc 函数等 The basic signals frequently used:complex sine signal,function,sinc function(sample function)etc.（4）时间函数信号的频谱密度-傅立叶变换 The spectrum density of time function signals-Fourier Transform 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)When complex sine signals are used as basic signals,a time functionsignal can be written with the form of Inverse Fourier Transform as deStstj)(21)(dtetsStj)()()(S)(ts)(ts)(ts)(SdeStstj)(21)()()(StsWhere is called the spectrum density,or the Fourier Transform of the ,andthe is the Inverse Fourier Transform of the .This relation is denoted as .The two functions are called a FT pair.2、傅立叶变换的重要特性 The important properties of FT（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。)(1ts)(2ts)(tsndtetsStj)()()(ts)(ts)(S)()(Sts)(1S)(2S)(nSniiiniiiSatsa11)()(nia The following equality will hold if ,are thecorresponding Fourier transforms of ,respectively:Where is an integer and s are constant coefficients.（2）尺度性质 Scale transformation若 ,则对实常数 有 (1.2.25)If ,then for a real constant ,the following equality holds:)(1S)(2S)(nS)(1ts)(2ts)(tsnniiiniiiSatsa11)()(nia)()(Stsa)(1)(Saats)()(Stsa)(1)(Saats（3）时延性质 Time Delay若 ,则有 (1.2.26)If ,then the following equality holds:（4）频移性质 Frequency Shift若 ,则有 (1.2.27)If ,then the following equality holds:（5）时域微分与积分 Differential and integral in time domain 若 ,则下列各式成立 If ,then following equalities hold:)()(00 Setstj)()(Sts)()(Sts0)()(0tjeStts)()(Sts0)()(0tjeStts)()(Sts)()(00 Setstj)()(Sts)()(Sts (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 Time Reverse 若 ,则有 (1.2.32)（7）对偶性 Duality 若 ,则有 (1.2.33)()(Sjdttds)()()(Sjdttsdnnn)()0(21)(1)(SSjdttsttSjdtts)(1)()(ts),()()(Sts)()(Sts)()(Sts)(2)(stS（8）时域卷积 Time domain convolution 若 ,则有 (1.2.34)（9）频域卷积 Frequency domain convolution 若 ,，则有 （1.2.35-1）或记为 (1.2.35-2)（10）复共轭特性 Complex conjugation 若 ,则有 (1.2.36)(1.2.37)()()()(2121SSdtsts)()(11Sts)()(22Sts)()(StsdSStsts)()()()(2121)(*)()()(2121SStsts)()(Sts)()(*Sts)()(*Sts3、典型函数的傅立叶变换 The FT of typical functions（1）单位脉冲函数（函数）Unit pulse function(function)Definition:(1.2.38)moreover Feature:FT:(1.2.39)or denoted as Inverse FT:000)(ttt1)(dtt)0()()(fdtttf1)()(0ttjtjedtet1)(t)(21t（2）单位阶跃函数 Unit jump function Definition:(1.2.40)FT:(1.2.41)（3）指数函数 Exponential function （1.2.42）Prove:According to the frequency shifting feature and We have 0001)(tttI)(211)(jtI)(200tje)()(00 Setstj)(21)(2100tje（4）正弦与余弦函数 Sine and Cosine functions (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 The rectangular pulse function with the amplitude width T¢er at origin i.e.(1.2.45)(21)cos(000tjtjeet)()()cos(000t)()()sin(000jt其它20)()(TtATtrectAts)2(sin2)2sin()(22TcATTTATdtAeSTTtj)2(sin)(TcATTtrectA2 相关函数和功率 The Correlation Functions&Power1、相关函数的普遍定义（应以遍历过程为条件）The general definition of correlation function(condition:ergodic process)自相关函数 The self-correlation function (1.2.46)互相关函数 The mutual correlation function (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。dttstsdttstsR)()()()()(*dttstsdttstsR)()()()()(*12*1212 Physical meaning:Describing the extent of the overlapping(correlated)between two signals:the integral will be zero when the two signals are notoverlapped thoroughly;maximum when they are overlapped thoroughly;between zero and maximum when they are overlapped partly.2、相关函数的傅立叶变换 The FT of correlation functions 互相关函数的傅立叶变换 The FT of self-correlation functions (1.2.48)Deriving:)()()(*xyxySSRdttstsRxyxy)()()(*dtdeStstjyx)(*)()(21dedtSetsjytjx)()(21*deSSjyx)()(21*If define then we have (1.2.49)巴塞瓦公式 Parseval Formula When ,formula(1.2.48-2)becomes (1.2.50)This is called Parseval Formula,which is the measurement of theextent of correlation of two signals in frequency domain.自相关函数的傅立叶变换及其能谱密度函数 The FT and the Energy Spectrum Density Functions of self-correlation functions )()()(*xyxySSRdttstsRyxxy)()()(*)()()(*yxxySSR0dSSRyxxy)()(21)0(*Substituting the subscript y by x in Form.（1.2.49）,we have (1.2.51)Therefore we can denote (1.2.52)where the is called the Energy Spectrum Density(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。Physical meaning:The self-correlation function and the energy spectrum density function compose a Fourier transform pair.deSSRjxxxx)()(21)(*deSjx2)(212*)()()()(xxxxxSSSR2)(xS)(ts3、能量型复信号和实信号的能量公式 The Energy Formula of energy-typed complex and real signals 复信号的能量公式：The Energy Formula of complex signals When ,according to the definition of self-correlation function and Form.（1.2.51）,we have (1.2.53)Formula(1.2.53)is called the Energy Formula of complex signals(复信号的能量公式）.0dttsdttstsR2*)()()()0(dS2)(21 实信号的能量公式：The Energy Formula of real signals When ,according to the definition of self-correlation function and Form.（1.2.51）,we have (1.2.54)Formula(1.2.54)is called the Energy Formula of real signals.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Physical meaning:The left side of the equality sign is the integral of signal power in time domain,i.e.the energy of the signal;the right side is the integralof the square of the modulus of the frequency spectrum of the signal in freq.domain,which is also the energy.Therefore the square of the modulus of thefrequency spectrum is called as the Energy Spectrum Density of the signal.dttsR)()0(2dSx2)(2104、功率型信号的相关函数与功率谱 The correlation functions&power spectra of power-typed signals （1）两种类型的信号 Two kinds of signals 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-typed signals:The energy of a signal is finite and the average power approaches to zero in the existing time of it.功率型信号：在 区间内信号功率有限而能量无 限的信号。Power-typed signals:The power of a signal is finite and the energy is infinite in the existing time of it.tttt（2）功率型信号的自相关函数 The self-correlation function of power-typed signals (1.2.55)Physical meaning:The average power of the signal (信号的平均功率)（3）功率型信号的傅立叶变换；功率谱密度 The FT of power-typed signals:Power Spectrum Density (1.2.56-1)dttstsRTTs)()(lim)(1)()(211lim)(*deSSTRjTsdeSTjT2)(1lim21dePjs)(21i.e.(1.2.56-2)Where the is called the Power Spectrum Density of signal .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:The self-correlation function and power spectrum densityFunction of a power-typed signal compose a FT pair.(4)功率型信号的平均功率 The average power of a power-typed signal let in Form.(1.2.56-1),we have (1.2.57)()(1lim)(2sTspSTR)(sp)(ts0dpRPss)(21)0(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:其它TttstsT00)()()(TS)(tsT)()(TTStsdttstsRTTTsT)()()(1)(lim)(1sTTTsRR)(ts)(S2)(1lim)(STPTs 对功率型信号有 (1.2.60)故有 (1.2.61)It should be noticed that in the procedure of solving ,It is impossible to obtain an exact frequency spectrum function ,because that the energy of is infinite.Therefore,a truncated functionlasting a finite piece of time should be defined,in advance,asThe is a time finite function,which has an exact frequency spectrumfunction ,and the relation:Let 2)()(SR2)(1)(TsTSTR)()(1lim)(2sTsPSTR2)(1lim)(STPTs)(S)(ts其它TttstsT00)()()(tsT)(TS)()(TTStsdttstsRTTTsT)()()(1We haveOn the analogy of the relation between the self-correlation and energy spectrum density of the energy-typed signals:we have for power-typed signals.Let and solve the limit of above formula,then we have the conclusion:5、随机过程样本函数的功率谱 The power spectrum of the sample functions of RPs 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)(tx)(TX2)(TX)(lim)(1sTTTsRR2)()(SR2)(1)(TsTSTRT)()(1lim)(2sTsPSTR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。The sample function of a RP is power-typed.Considering therandom property of its truncated frequency spectrum ,the statisticalaveraging of should be carried when solving its power spectrum.Therefore the formula of the power spectrum density of a RP becomes where The“”expresses statistical average and the can be neglected whenthe magnitude of is not interesting.T12)(1lim)(TTsXTp)()(sspR)(tx)(TX2)(TX)()(sspR2)(1lim)(TTsXTpT1)(sp6、小结 Summary:Important relations:频谱密度 The Frequency Spectrum Density Function：能量谱密度 The Energy Spectrum Density Function：能量公式 The Energy Formula(Parseval Formula in other books):功率谱密度 The Power Spectrum Density Function：平均功率 The Average Power of a RP:)()(Sts2)()(SRdSRx2)(21)0()()(1lim)(2sTspSTRdpRPss)(21)0(