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1、Chapter 5 Time delay Correlation Analysis Fundamentals of Measurement and Signal Analysis 5.2 Properties of correlation function 1)Auto-correlation function:The auto-correlation function is an even function，Rx()=Rx(-);When =0，the auto-correlation function has the maximum value;The auto-correlation f
2、unction of a periodic signal is still a periodic signal of the same frequency,but it does not retain the phase information of the original signal;The auto-correlation function of the random noise signal will decay rapidly as increases.()()()xRx t x tdtPositive time delay Negative time delay 0 The au
3、tocorrelation function is an even function，Rx()=Rx(-);When =0，the autocorrelation function has the maximum value;x t()xR()5.2 Properties of correlation function The autocorrelation function of a periodic signal is still a periodic signal of the same frequency,but it does not retain the phase informa
4、tion of the original signal;Fs=500;dt=1.0/Fs;P=0;T=1;N=T/dt;x=linspace(0,T,N);for i=1:50 y=2*sin(2*3.14*5*x+P*pi/180.0);subplot(2,1,1);plot(x,y,b,linewidth,1);xlim(0,T);ylim(-2,2);grid on;s1=xcorr(y,unbiased);N1=length(s1);x1=linspace(-T,T,N1);subplot(2,1,2);plot(x1,s1,b,linewidth,1);xlim(-T/2,T/2);
5、ylim(-2,2);grid on;P=P+10;pause(1);end Positive time delay Negative time delay 0 x t()xR()5.2 Properties of correlation function The autocorrelation function of the random noise signal will decay rapidly as increases.Fs=500;dt=1.0/Fs;T=1;N=T/dt;x=linspace(0,T,N);for i=1:50 y=randn(1,N);subplot(2,1,1
6、);plot(x,y,b,linewidth,1);xlim(0,T);ylim(-3,3);grid on;s1=xcorr(y,unbiased);N1=length(s1);x1=linspace(-T,T,N1);subplot(2,1,2);plot(x1,s1,b,linewidth,1);xlim(-T/2,T/2);ylim(-1,1);grid on;pause(1);end Positive time delay Negative time delay 0 x t()xR()5.2 Properties of correlation function Experiment：
7、Auto-correlation function of standard signals 5.2 Properties of correlation function 2)Cross-correlation function:The cross-correlation function of two periodic signals is still a periodic signal of the same frequency,and the phase information of the original signal is retained.Two periodic signals
8、with different frequencies are not correlated to each other.()()()xyRx t y tdt5.2 Properties of correlation function The cross-correlation function of two periodic signals is still a periodic signal of the same frequency,and the phase information of the original signal is retained.Fs=500;dt=1.0/Fs;P
9、=0;T=1;N=T/dt;x=linspace(0,T,N);y1=2*sin(2*3.14*5*x);for i=1:50 y2=2*sin(2*3.14*5*x+P*pi/180.0);subplot(3,1,1);plot(x,y1,b,linewidth,1);xlim(0,T);ylim(-2,2);grid on;subplot(3,1,2);plot(x,y2,b,linewidth,1);xlim(0,T);ylim(-2,2);grid on;s1=xcorr(y1,y2,unbiased);N1=length(s1);x1=linspace(-T,T,N1);subplo
10、t(3,1,3);plot(x1,s1,b,linewidth,1);xlim(-T/2,T/2);ylim(-2,2);grid on;P=P+10;pause(1);end x t()x t()y t()xyR()Positive time delay Negative time delay 0 5.2 Properties of correlation function Fs=500;dt=1.0/Fs;F=5;T=1;N=T/dt;x=linspace(0,T,N);y1=2*sin(2*3.14*5*x);for i=1:20 y2=2*sin(2*3.14*F*x);F=F+2;s
11、ubplot(3,1,1);plot(x,y1,b,linewidth,1);xlim(0,T);ylim(-2,2);grid on;subplot(3,1,2);plot(x,y2,b,linewidth,1);xlim(0,T);ylim(-2,2);grid on;s1=xcorr(y1,y2,unbiased);N1=length(s1);x1=linspace(-T,T,N1);subplot(3,1,3);plot(x1,s1,b,linewidth,1);xlim(-T/2,T/2);ylim(-2,2);grid on;pause(1);end Two periodic si
12、gnals with different frequencies are not correlated to each other.x t()y t()xyR()Positive time delay Negative time delay 0 5.2 Properties of correlation function Experiment：Cross-correlation function of sine signals Positive time delay Negative time delay 0 5.2 Properties of correlation function 3)C
13、onvolution,Correlation and Fourier Transform Convolution is a mathematical operation used to express the relationship between input and output of an system.()()*()()()y tx th txh td Correlation is a measure of similarity between two signals.xhRtxht dx tht()()()()*()()()()xyRx t y tdt(Write in the sa
14、me form)5.2 Properties of correlation function The Fourier transform of the convolution is the product of the two Fourier transforms.The Fourier transform of the correlation is the product of the Fourier transform of one function by the complex conjugate of the Fourier transform of another.y tx th t
15、Y fX f H f()()*()()()()xyRcorr x yP fX f Yf*()(,)()()()5.2 Properties of correlation function y tX f H f-1()F()()（）y tX ff-1*()F()Y()（）Example:Calculation of Convolution and Correlation x t()h t()x th t()*()corr x h(,)R1=ovelap area 0 0 1 1 1 1 0 0 0 0 4 3 2 1 0 0 5.2 Properties of correlation funct
16、ion V1=ovelap area Demo Example:Calculation of Convolution and Correlation x t()h t()x th t()*()corr x h(,)0 0 1 1 1 1 0 0 0 0 4 3 2 1 0 0 5.2 Properties of correlation function Homeworks 1.What are the main properties of correlation analysis?5.2 Properties of correlation function Fundamentals of Measurement and Signal Analysis