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1、 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 functionsignal can be written with the form of Inverse Fourier Transform as Where is called the spectrum density, or the Fourier Transform of the , andthe is the
2、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 thecorresponding Fourier transfor
3、ms 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 , t
4、hen 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
5、)（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)moreoverFeature: FT:(1.2.39)or denoted as Inverse FT: （2） 单位阶跃函数 Unit jump f
6、unctionDefinition: (1.2.40)FT: (1.2.41)（3）指数函数 Exponential function（1.2.42）Prove: According to the frequency shifting featureand 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
7、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 FormulaWhen , formula (1.2.48-2) becomes(1.2.50) This is ca
8、lled 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-correlationfunctions Substituting the subscript y by x in Form.（1.2.49）, we have(1.2.51)Therefore we can denote (1
9、.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 spectrumdensity 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
10、能量公式：The Energy Formula of complex signalsWhen , 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 s
11、elf-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
