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1、 2.2 随机过程自相关函数与功率谱随机过程自相关函数与功率谱 The self-correlation functions&power spectra of RPs 一信号频谱和傅立叶变换 The spectra&Fourier transforms of signals1、基本概念 Basic concepts（1）实信号：可用时间实函数表示信号 Real signal:The signals expressed with real function of time are.特点：含有有限能量或有限功率 Features:The energy or power of a real sign
2、al is finite.（2）能量信号：能量有限信号 Energy signal:The signals with finite energies下一页第1页 （3）时间函数信号分解 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总和或积分 A time function signal can be expressed with the sum or integral of a certain number of basic signals 惯用基本信号：复正弦信号、函数、sinc 函数等 The basic si
3、gnals frequently used:complex sine signal,function,sinc function(sample function)etc.（4）时间函数信号频谱密度-傅立叶变换 The spectrum density of time function signals-Fourier Transform第2页 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)When complex sine signals are
4、used as basic signals,a time functionsignal can be written with the form of Inverse Fourier Transform as 第3页Where 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
5、换主要特征 The important properties of FT（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。第4页 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 (1.2.2
6、5)If ,then for a real constant ,the following equality holds:第5页（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 ,则以下各式成立 If ,then following equalitie
7、s hold:第6页 (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)第7页（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）复共轭特征 Complex conjugation 若 ,则有 (1.2.36)(1.2
8、.37)第8页3、经典函数傅立叶变换 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:第9页（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
9、 feature and We have 第10页（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、中心位于原点矩形脉冲函数 The rectangular pulse function with the amplitude width T¢er at origin i.e.(1.2.45)第11页二相关函数和功率 The Correlation Functions&Power1、相关函数普遍定义（应以遍历过程为条件）The general definition of correlati
10、on function(condition:ergodic process)自相关函数 The self-correlation function (1.2.46)相互关函数 The mutual correlation function (1.2.47)物理含义:两个信号之间交迭程度(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。第12页 Physical meaning:Describing the extent of the overlapping(correlated)between two signals:the integral
11、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、相关函数傅立叶变换 The FT of correlation functions 相互关函数傅立叶变换 The FT of self-correlation functions (1.2.48)Deriving:第13页If define then we have
12、(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 第14页Substitu
13、ting 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:能量信号自相关函数与能量谱密度函数组成傅立叶 变换对。Physical meaning:The self-correlation function and the energy spectrum density function compose a Fourier transform pair
14、.第15页3、能量型复信号和实信号能量公式 The Energy Formula of energy-typed complex and real signals 复信号能量公式：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(复信号能量公式）.第16
15、页 实信号能量公式：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
16、 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 Spectr
17、um Density of the signal.第17页4、功率型信号相关函数与功率谱 The correlation functions&power spectra of power-typed signals （1）两种类型信号 Two kinds of signals 能量型信号：在整个信号存在时间 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
18、型信号：在 区间内信号功率有限而能量无 限信号。Power-typed signals:The power of a signal is finite and the energy is infinite in the existing time of it.第18页（2）功率型信号自相关函数 The self-correlation function of power-typed signals (1.2.55)Physical meaning:The average power of the signal (信号平均功率)（3）功率型信号傅立叶变换；功率谱密度 The FT of powe
19、r-typed signals:Power Spectrum Density (1.2.56-1)第19页i.e.(1.2.56-2)Where the is called the Power Spectrum Density of signal .上式表明:功率型信号自相关函数与其功率谱密度函数组成傅立叶变换对。Meaning:The self-correlation function and power spectrum densityFunction of a power-typed signal compose a FT pair.(4)功率型信号平均功率 The average po
20、wer of a power-typed signal let in Form.(1.2.56-1),we have (1.2.57)第20页注意到：求 过程中因为 是能量无限，故不可能求出确切频谱 ，为此须先定义一个连续时间有限截短函数 (1.2.58)使得 成为能量有限函数,可有确切频谱 ,即有 令 则有 (1.2.59)类比于能量型信号关系:第21页 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 fr
21、equency 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 第22页We haveOn the analogy of the relation be
22、tween 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,但考虑到其频谱随机性,在求其功率谱时还须对 作统计平均,故随机过程第23
23、页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 becom
24、es where The“”expresses statistical average and the can be neglected whenthe magnitude of is not interesting.第24页6、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:第25页