8: Correlation - Imperial College London

1 8: Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 1 / 11 Cross-Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d Cross-Correlation8.

2 Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub: t]Cross-Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub: t]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is.

3 Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub: t]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is versus Convolution:u(t) v(t) = u ( )v( +t)d [ Correlation ]u(t) v(t) = u( )v(t )d [convolution]Cross-Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub.]

4 T]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is versus Convolution:u(t) v(t) = u ( )v( +t)d [ Correlation ]u(t) v(t) = u( )v(t )d [convolution]Unlike convolution, the integration variable, , has thesame signin thearguments ofu( )andv( )so the arguments have aconstantdifferenceinstead of a constant sum ( (t)is not time-flipped).Cross-Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub: t]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is versus Convolution.

5 U(t) v(t) = u ( )v( +t)d [ Correlation ]u(t) v(t) = u( )v(t )d [convolution]Unlike convolution, the integration variable, , has thesame signin thearguments ofu( )andv( )so the arguments have aconstantdifferenceinstead of a constant sum ( (t)is not time-flipped).Notes:(a) The argument ofw(t)is called the lag (= delay ofuversusv).Cross-Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub: t]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is versus Convolution.

6 U(t) v(t) = u ( )v( +t)d [ Correlation ]u(t) v(t) = u( )v(t )d [convolution]Unlike convolution, the integration variable, , has thesame signin thearguments ofu( )andv( )so the arguments have aconstantdifferenceinstead of a constant sum ( (t)is not time-flipped).Notes:(a) The argument ofw(t)is called the lag (= delay ofuversusv).(b) Some people writeu(t) v(t)instead ofu(t) v(t).Cross-Correlation8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub: t]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is versus Convolution.

7 U(t) v(t) = u ( )v( +t)d [ Correlation ]u(t) v(t) = u( )v(t )d [convolution]Unlike convolution, the integration variable, , has thesame signin thearguments ofu( )andv( )so the arguments have aconstantdifferenceinstead of a constant sum ( (t)is not time-flipped).Notes:(a) The argument ofw(t)is called the lag (= delay ofuversusv).(b) Some people writeu(t) v(t)instead ofu(t) v(t).(c) Some swapuandvand/or negatetin the : Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 2 / 11 Thecross-correlationbetween two signalsu(t)andv(t)isw(t) =u(t) v(t), u ( )v( +t)d = u ( t)v( )d [sub.]

8 T]The complex conjugate,u ( )makes no difference ifu(t)is real-valuedbut makes the definition work even ifu(t)is versus Convolution:u(t) v(t) = u ( )v( +t)d [ Correlation ]u(t) v(t) = u( )v(t )d [convolution]Unlike convolution, the integration variable, , has thesame signin thearguments ofu( )andv( )so the arguments have aconstantdifferenceinstead of a constant sum ( (t)is not time-flipped).Notes:(a) The argument ofw(t)is called the lag (= delay ofuversusv).(b) Some people writeu(t) v(t)instead ofu(t) v(t).(c) Some swapuandvand/or negatetin the is all rather inconsistent/.

9 Signal Matching8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 3 / 11 Cross Correlation is used to find where twosignals match0000000000000000000000000000000000 Signal Matching8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 3 / 11 Cross Correlation is used to find where twosignals match:u(t)is the test (t)000000000000000000000000000000 Signal Matching8.

10 Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation : 8 3 / 11 Cross Correlation is used to find where twosignals match:u(t)is the test 1:v(t)containsu(t)with an unknown delayand added (t) (t)0000000000000000000000000 Signal Matching8: Correlation Cross- Correlation Signal Matching Cross-corr as Convolution Normalized Cross-corr Autocorrelation Autocorrelation example Fourier Transform Variants Scale Factors Summary Fourier Series and Transforms (2015-5585)Fourier Transform - Correlation .