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Session 2 : Correlation

Correlation

Ir. Dadang Gunawan, Ph.DElectrical Engineering

University of Indonesia


The Outline2.1 State-of- the-art2.2 Philosophy of Correlation2.3 Auto and Cross Correlation2.4 Example of Correlation (j=0)2.5 Correlation with phase differences2.6 Example of Correlation (lag j=3)2.7 Auto correlation2.8 Cyclic-cross Correlation

(Cont’d… )


The Outline

2.9 Example of Cyclic-cross Correlation2.10 Example of Graphic Correlation2.11 Cross-correlation of two noisy waveform2.12 Applications of Correlation2.13 Review


State of the art

• Correlation is one of the key of DSP• Require simple arithmetic operations• The purpose :To determine how is the correlation between two

signals in a systems.Correlation is used together with convolutionIf two signals are the same, the value of its

correlation is 1 ( one ), look at the equation on next page

2.1


Philosophy of Correlation

• Correlation is an integral part of convolution• Correlations of two waveforms is the sum of the

products of the corresponding pairs of points• Note that x1(n) and x2(n) are the corresponding

pairs of 2 waveforms

∑−

=

=1

02112 )()(1 N

nnxnx

Nr

2.2


Auto and Cross-correlation

2.3

Auto Correlation

Cross Correlation

Correlation with himself

Correlation with other waveform


Example of Correlation (j=0)

n 1 2 3 4 5 6 7 8 9

x1 4 2 -1 3 -2 -6 -5 4 5x2 -4 1 3 7 4 -2 -8 -2 1

( ) ( ) ( ) ( )[ ]( ) 50

15...1244910

12

12

=

×++×+−×=

r

r

2.4


Correlation with phase differences (shift or lag one of the waveforms)

• Look at the figure 2.1 below• This is equivalent to changing x1(n) to x2(n+j), where

j represents the amount of lag, shifted to the left

2.5

Figure 2.1


Alternative shifted to right

• It is equivalent,but different in direction• The formula a lag of j , are :

∑

∑−

=

−

=

−=−

+=

1

01221

1

02112

)()(1)(

)()(1)(

N

n

N

n

jnxnxN

jr

jnxnxN

jr

2.5


Example of Correlation ( lag of j=3)

n 1 2 3 4 5 6 7 8 9

x1 4 2 -1 3 -2 -6 -5 4 5x2 7 4 -2 -8 -2 1

( ) ( ) ( ) ( )[ ]( ) 667.23

16...4274913

12

12

=

×−++×+×=

r

r

2.6


Auto correlation

• Occur when x1(n)=x2(n), and the waveform is then cross-correlated with itself

• The formula is :

• It has a property :

( ) ∑−

=

+=1

01111 )()(1 N

njnxnx

Njr

( ) ∑−

=

==1

0

2111 )(10

N

nSnx

Nr

Remember that ( ) ( )jrr 1111 0 ≥

2.7


Autocorrelation (cont’d)• That is a very useful property that S is the

normalized energy of the waveform• This provides a method for calculating the energy

of the signal, look at the figure 2.2 below :

2.7

j

r11

Figure 2.2


Cyclic-cross correlation

• rab (j) is cyclic, repeating every n lag• For example, if the sequence of a is 4 and b is 3,

then n = 4 + 3 – 1 = 6• So that, n = N1 + N2 - 1• That is rab (j) has the same period as that of the

shorter sequence b• Look at the example on the next slide

2.8


Example of cyclic cross-correlation

Sequence Lag rab(j)4 3 1 6 0 05 2 3 0 0 0 0 292 3 0 0 0 5 1 173 0 0 0 5 2 2 120 0 0 5 2 3 3 300 0 5 2 3 0 4 170 5 2 3 0 0 5 355 2 3 0 0 0 6 29

etc etc etc

rab(j)repeat

a

b

2.9


Example of Graphic correlation

• Look at the figure 2.3 below, the waveform v1(t)and v2(t)

2.10

v1(t) v2(t)Figure 2.3


Graphic correlation (cont’d)

• Here is the cross-correlation r12(-τ) between two waveforms (previous figure) on period 0-T

2.10

Figure 2.4


Cross Correlation of two noisy waveforms

• Consider two waveforms be {s1(t) + q1(t)} and {s2(t) + q2(t) }

• Then the correlation become :

• Look at each component, it is very useful to determine the noisy signal

( ) ( ) ( ) ( ) ( )jrjrjrjrjr qqsqqsss 2121212112 +++=

2.11


The autocorrelation function of noisy signal (related to previous slide)

2.11

2sq+q2

Figure 2.5

rvv(j)


Applications of Correlation

1. Calculations of energy spectral density and energy content of waveforms

The total energy of two waveform v1(n) and v2(n)are:

Remember that ( ) Er =011

( ) ( ) ( )0200 2121 vvvvv rrrE ++=

2.12


Applications of Correlation (cont’d)

2. Detection and estimation of periodic signals in noise.It will be done by making an adjustable template signal

2.12

Explain how can be done ?


Applications of Correlation (cont’d)

3. Correlation detection implementation of matched filter

4. The determination of the impulse response of electrical systems

5. Determination of the SNR for a periodic noisy signal

2.12

Explain how can be done ?


Special Review

• An application of correlation is to control attitude of a spacecraft, to ensure that the solar panel always faces the sun.

2.13

EXPLAIN CLEARLYand describe its graphic !


Review

1. Do the special review on previous slides.2. Explain about end-effect clearly.3. Explain about correlation detection implemen-

tation of matched filter. Look at other reference: internet, journal, etc.

4. The exercise from text book [Ifeachor] is postponed. You will do after Session 3, Convolution.

2.13

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