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 The self-correlation functions 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 function signal can be written with the form of Inverse Fourier Transform as Where is called the spectrum density, or the Fourier Transform of the , and the 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。 The following equality will hold if , , are the corresponding 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: （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: (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) （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) 3、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: （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 （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 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: 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 the extent 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 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. 3、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）. 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 integral of 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 the frequency spectrum is called as the Energy Spectrum Density of the signal. 4、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. （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) 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 density Function 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) 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) 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 function lasting a finite piece of time should be defined, in advance, as The is a time finite function, which has an exact frequency spectrum function , and the relation: Let We have On 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 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 functi