Correlation of Discrete Time Signals

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 

n 

l  0, 1, 2,



rxy  l    x n  l  y n

n 

l  0, 1, 2,



• Reversingthe order, ryx(l)

ryx  l    y n x n  l 

n 

l  0, 1, 2,



ryx  l    y n  l  x n

n 

l  0, 1, 2,



• => rxy l   ryx  l 

Similarity to Convolution

• No folding (time-reversal)



rxy  l   x l   y  l  ryx l   y l   x  l 



• In Matlab:

– Conv(x,fliplr(y))

Auto-Correlation

• Correlation of a signal with itself



rxx  l    x  n  x  n  l   r  l 

n 

xx l  0, 1, 2,





• 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)





 ax  n   by  n  l 

2

Ez   

n 

  

a 2

 x  n   b  y  n  l   2ab  x  n  y  n  l 

n 

2 2



n 

2



n 



 a 2 rxx  0   b 2 ryy  0   2abrxy  l   0

Ex Ey

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   Ex Ey



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]:



rxy  l  rxx  l 

 xx  l    xx  l  

rxx  0  ryy  0  rxx  0 

x  n   a nu  n  ; 0  a 1











rxx  l    x n x n  l 

n 

 

 a a n n l

a l

a 2n



n l n l



1

 l0

l

a

1  a2

rxx  l 

 xx  l   a

l

  l  

rxx  0 

Periodic Sequences

• Power signals crosscorrelation:

M

1

r  l   lim

xy

M  2M  1

n  M

 x n y n  l 

• Define auto and crosscorrelations over

one period of the signals

• If x(n) and y(n) are periodic signals with

period N: N 1

1

xy r l    x  n  y  n  l 

n 0N



• 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    r k  r k 

k 

hh xx