Presented By-

Er. Sanyam S. SainiME (I&CE) (Regular)

2012-14

Presented To-

Dr. Lini MathewAssociate Prof. (Electrical Deptt.)

NITTTR, Chandigarh

Correlation of Discrete-Time Signals

• Correlation gives a measure of similarity between two data sequence.

• Correlation is a comparison process.

• The correlation between two functions is a measure of their similarity.

• Correlation techniques are widely used in signal processing with

many applications in telecommunications, radar, medical

electronics, physics, astronomy, geophysics, fingerprint matching

etc.

Radar Target Detection

Reflected Signal, y(n)

y(n) = αx(n-D) + w(n)Here,y(n) = Sampled version of Received signalx(n) = Sampled version of transmitted signalw(n) = Noise that picked up by Antenna & noise

generated by electronic comp. & amp. In front of radar. (Additive Noise)

D = Round Trip Delayα = Attenuation factor (Loss in round trip transmission of x(n) )

Properties of Correlation

Detect wanted signal in the presence of noise or other unwanted signals.

1.

2.

3.

Example free space, various materials, solids, liquids, gases etc .

Recognise patterns within analogue, discrete-time or digital signals.

Allow the determination of time delays through various media.

Cross correlation Sequences

• In cross correlation, two ‘separate’ signals are compared.

.......3,2,1,0l

lnynxrn

xy

.......3,2,1,0l

nylnxrn

xyOr

If, we reverse the order of x(n) & y(n)

.......3,2,1,0l

lnxnyrn

yx

.......3,2,1,0l

nxlnyrn

yxOr

..........(i)

..........(ii)

On comparison

lrlr yxxy

Numerical on Cross correlation

Determine the cross correlation sequence of the following,

1,0,0,1nx 1,2,3,4ny&

Solution: lnynxrn

xy

Sr . l=0,±1, ±2,

1. l= 0 5

2. l= ±123

3. l= ±232

4. l= ±341

nynxrn

xy 0

Expression for rxy (l)

11 nynxrn

xy

rxy (l)

22 nynxrn

xy

33 nynxrn

xy

Auto correlation Sequences

When y(n) = x(n) , the cross correlation function become auto correlation function

.......3,2,1,0l

lnynxrn

xy

We know that

if y(n) = x(n)

therefore

.......3,2,1,0l

lnxnxrn

xx

.......3,2,1,0l

nylnxrn

xyOr

.......3,2,1,0l

nxlnxrn

xxOr

Auto correlation Sequences

In dealing with finite duration sequences, it is necessary to express the auto-correlation & cross correlation in terms of the finite limits on the summation

The correlation & auto correlation may be expressed as:

lnynxrkN

ln

xy

1

lnxnxlrkN

in

xx

1

If , x(n) & y(n) are causal sequences of length ‘N’ (i.e., x(n)=y(n)=0 for n<0 &n>N).

i=l , k=0 for l>=0

Where,

i=0 , k=l for l<0&

Compute the auto correlation of the signal

10, anuanx n

Since x(n) is an infinite- duration signal, its auto correlation also has infiniteduration.

solution

Considering two cases,

Numerical on Auto correlation

If , l>=0

0 0

2

0 n n

nllnn

n

xx aaaalnxnxlr

l

xx aa

lr21

1Hence

Numerical on Auto correlationx(n)

n

x(n-l)

nl

1

0-2 -1 0 1 2 ..

1

If , l<0

on

lnl

n

xx aa

aalnxnxlr2

2

0 1

1

x(n-l)

n l

1

-2 -1 0 1 2 . . .l o

1l<0

l>=0

l

xx aa

lr21

1

We can observe that, lrlr xxxx

Correlation of Periodic Sequences

Let x(n) & y(n) be two periodic signals.

1

0

1 N

n

xy lnynxN

lr

Their correlation sequences is defined as ,

if, x(n) =y(n)

1

0

1 N

n

xx lnxnxN

lr

Application of Correlation

Radar Target Detection

Thank You