1. Presented To- Presented By-
Dr. Lini Mathew Er. Sanyam S. Saini
Associate Prof. (Electrical Deptt.) ME (I&CE) (Regular)
NITTTR, Chandigarh 2012-14

2. 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.

3. Radar Target Detection
y(n) = αx(n-D) + w(n)
Here,
y(n) = Sampled version of Received signal
x(n) = Sampled version of transmitted signal
w(n) = Noise that picked up by Antenna & noise
generated by electronic comp. & amp. In front
of radar. (Additive Noise)
D = Round Trip Delay
α = Attenuation factor (Loss in round trip
transmission of x(n) )
Reflected Signal, y(n)

4. Properties of Correlation
Detect wanted signal in the presence of noise or other unwanted
1. signals.
2. Recognise patterns within analogue, discrete-time or digital
signals.
3. Allow the determination of time delays through various media.
Example free space, various materials, solids, liquids, gases etc .

5. Cross correlation Sequences
• In cross correlation, two ‘separate’ signals are compared.
rxy xnyn l rxy xn l yn
Or
n n
l 0, 1, 2, 3....... l 0, 1, 2, 3....... ..........(i)
If, we reverse the order of x(n) & y(n)
ryx ynxn l ryx yn lxn
Or
n n
l 0, 1, 2, 3....... l 0, 1, 2, 3....... ..........(ii)
On comparison
rxy l ryx l

6. Numerical on Cross correlation
Determine the cross correlation sequence of the following,
x n 1,0,0,1 & y n 4,3,2,1
Solution: rxy x n y n l
n
Sr . l=0,±1, ±2, Expression for rxy (l) rxy (l)
1. l= 0 rxy 0 x n y n 5
n
rxy 1 x n y n 1 2
2. l= ±1 3
n
3. 3
l= ±2 rxy 2 x n y n2
n
2
4. rxy 3 x n y n3 4
l= ±3
n 1

7. Auto correlation Sequences
When y(n) = x(n) , the cross correlation function become auto correlation function
We know that
rxy x n y n l rxy xn l yn
n
Or
n
l 0, 1, 2, 3....... l 0, 1, 2, 3.......
if y(n) = x(n)
therefore
rxx xnxn l rxx xn l xn
n
Or n
l 0, 1, 2, 3....... l 0, 1, 2, 3.......

8. 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
If , x(n) & y(n) are causal sequences of length ‘N’ (i.e., x(n)=y(n)=0 for n<0 &
n>N).
The correlation & auto correlation may be expressed as:
N k 1 N k 1
rxy xnyn l rxx l xnxn l
n l n i
Where,
i=l , k=0 for l>=0 & i=0 , k=l for l<0

9. Numerical on Auto correlation
Compute the auto correlation of the signal
xn a nu n ,0 a 1
solution
Since x(n) is an infinite- duration signal, its auto correlation also has infinite
duration.
Considering two cases,
n n l l 2 n
If , l>=0 rxx l xnxn l a a a a
n 0 n 0 n 0
1 l
Hence rxx l 2
a
1 a

10. Numerical on Auto correlation
x(n) x(n-l)
1 1 l>=0
n n
-2 -1 0 1 2 .. 0 l
If , l<0 1
l 2 n l
rxx l xnxn l a a a
n 0 n o 1 a2
x(n-l) 1 l
rxx l a
1 1 1 a2
l<0
n l
l o -2 -1 0 1 2 . . .
We can observe that, rxx l rxx l

11. Correlation of Periodic Sequences
Let x(n) & y(n) be two periodic signals.
Their correlation sequences is defined as ,
N 1
1
rxy l xn yn l
N n 0
if, x(n) =y(n)
N 1
1
rxx l xnxn l
N n 0

12. Application of Correlation
Radar Target Detection

13. Thank You