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)

• Reversing the order, ryx(l)

• =>

0, 1, 2,

0, 1, 2,

xyn

xyn

r l x n y n l l

r l x n l y n l

0, 1, 2,

0, 1, 2,

yxn

yxn

r l y n x n l l

r l y n l x n l

xy yxr l r l

Similarity to Convolution

• No folding (time-reversal)

• In Matlab:– Conv(x,fliplr(y))

xyr l x l y l yxr l y l x l

Auto-Correlation

• Correlation of a signal with itself

• Used to differentiate the presence of a like-signal, e.g., zero or one

• Even function

0, 1, 2,xx xxn

r l x n x n l r l l

Properties

• Two sequences, x(n) and y(n), with finite energy

• Find energy of z(n)

z n ax n by n l

2

2 2 2 2

2 2

2

0 0 2 0

zn

n n n

xx yy xy

E ax n by n l

a x n b y n l ab x n y n l

a r b r abr l

Ex Ey

2 2

2

0 0 2 0 ( 0)

0 2 0 0

z xx yy xy

xx xy yy

E a r b r abr l assume b

a ar r l r

b b

Quadratric in (a/b) and positive, discriminant is non-negative:For crosscorrelation case:

For autocorrelation case:

0 0xy xx yy x yr l r r E E

0xx xx xr l r E

Maximum value occurs with zero lag (when signals are perfectly matched)Often normalized to range [-1,1]:

00 0

xy xxxx xx

xxxx yy

r l r ll l

rr r

; 0 1nx n a u n a

2

2

10

1

xxn

n n l l n

n l n l

l

r l x n x n l

a a a a

a la

0

lxxxx

xx

r ll a l

r

Periodic Sequences

• Power signals crosscorrelation:

• Define auto and crosscorrelations over one period of the signals

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

• Correlations are also periodic with period N

1lim

2 1

M

xyM

n M

r l x n y n lM

1

0

1 N

xyn

r l x n y n lN

y(n)=x(n)+w(n)

LTI Systems

• Convolution, output of LTI system

• Crosscorrelation between the output and input signal:

• Similarly, input to output is:

k

y n h n x n h k x n k

yx

xx

r l y l x l h l x l x l

h l x l x l

h l r l

xy xxr l h l r l

• Autocorrelation of output:

yy

hh xx

r l y l y l h l x l h l x l

h l h l x l x l

r l r l

0 0 0yy hh xx hh xxk

r r r r k r k