Correlation of Discrete-Time Signals - Iowa State University

Correlation of Discrete-Time Signals Transmitted Signal, x(n) Reflected Signal, y(n) = x(n-D) + w(n) 0 T Cross-Correlation Cross-correlation of x(n) and y(n) is a sequence, rxy(l) rxy l x n y n l l 0, 1, 2, n

rxy l x n l y n l 0, 1, 2, n Reversing the order, ryx(l) ryx l y n x n l l 0, 1, 2, n ryx l y n l x n n => rxy l ryx l l 0, 1, 2,

Similarity to Convolution No folding (time-reversal) rxy l x l y l In Matlab: Conv(x,fliplr(y)) ryx l y l x l Auto-Correlation Correlation of a signal with itself rxx l x n x n l rxx l l 0, 1, 2, n Used to differentiate the presence of a

like-signal, e.g., zero or one Even function Properties Two sequences, x(n) and y(n), with finite energy z n ax n by n l Find energy of z(n) 2 Ez ax n by n l n a 2 2 2

2 x n b y n l 2ab x n y n l n n a 2 rxx 0 b 2 ryy 0 2abrxy l 0 Ex Ey n Ez a 2 rxx 0 b 2 ryy 0 2abrxy l 0

(assume b 0) 2 a a rxx 0 2 rxy l ryy 0 0 b b Quadratric in (a/b) and positive, discriminant is non-negative: For crosscorrelation case: rxy l rxx 0 ryy 0 E x E y For autocorrelation case: rxx l rxx 0 Ex Maximum value occurs with zero lag (when signals are perfectly matched) Often normalized to range [-1,1]: xx l

rxy l rxx 0 ryy 0 xx l rxx l rxx 0 x n a nu n ; 0 a 1 rxx l x n x n l n n

a a n l n l 1 l a 1 a2 a l a

2n n l l 0 xx l rxx l l a rxx 0 l Periodic Sequences Power signals crosscorrelation: 1 r l lim x n y n l 2M 1

Define auto and crosscorrelations over one period of the signals If x(n) and y(n) are periodic signals with period N: 1 r l x n y n l N M xy M n M N1 xy n 0

Correlations are also periodic with period N y(n)=x(n)+w(n) LTI Systems Convolution, output of LTI system y n h n x n h k x n k k Crosscorrelation between the output and input signal: r l y l x l h l x l x l yx h l x l x l h l rxx l Similarly, input to output is: rxy l h l rxx l

Autocorrelation of output: ryy l y l y l h l x l h l x l h l h l x l x l rhh l rxx l ryy 0 rhh 0 rxx 0 rhh k rxx k k