correlation correlation

8/13/2019 Correlation correlation

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Signal Processing

Mike DoggettStaffordshire University


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CORRELATION

Introduction Correlation FunctionContinuous-Time

Functions

Auto Correlation and Cross Correlation

Functions

Correlation Coefficient

CorrelationDiscrete-Time Signals

Correlation of Digital Signals


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INTRODUCTION

Correlation techniques are widely used in signal

processing with many applications in

telecommunications, radar, medical electronics,

physics, astronomy, geophysics etc . . .. .


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Correlation has many useful properties, giving forexample the ability to:

Detect a wanted signal in the presence of noiseor other unwanted signals.

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

Allow the determination of time delays throughvarious media, eg free space, various materials,solids, liquids, gases etc . . .


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Correlation is a comparison process.

The correlation betweeen two functions is a

measure of their similarity.

The two functions could be very varied. Forexample fingerprints: a fingerprint expert can

measure the correlation between two sets of

fingerprints.


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This section will consider the correlation of

signals expressed as functions of time. The

signals could be continuous, discrete time ordigital.

When measuring the correlation between twofunctions, the result is often expressed as a

correlation coefficient, , with in the range1

to +1.


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Correlation involves multiplying, sliding and

integrating

Similar but opposite No similarity Exactly similar

= - 1 = 0 = +1


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Consider 2 functions


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Consider 2 more functions


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Consider 2 more functions


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CONVOLUTION

t = 0

0010110111

1110110100

SYSTEM

Signal


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CORRELATION FUNCTIONCONTINUOUS

TIME FUNCTIONS

Consider two continuous functions of time, v1(t)

and v2(t). The functions may be random or

deterministic.

The correlation or similarity between these twofunctions measured over the interval T is given

by:

2

2

21 )()(1

12

T

T

dttvtvT

T

R Lim


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The functions may be deterministic or random.

R12() is the correlation function and is ameasure of the similarity between the functions

v1(t) and v2(t).

The measure of correlation is a function of a new

variable, , which represents a time delay or time

shift between the two functions.


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Note that correlation is determined by multiplying

one signal, v1(t), by another signal shifted in time,

v2(t-), and then finding the integral of theproduct,

Thus correlation involves multiplication, timeshifting (or delay) and integration.


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The integral finds the average value of the

product of the two functions, averaged over a

long time (T ) for non-periodic functions.

For periodic functions, with period T, the

correlation function is given by:

2

2

21 )()(1

12

T

T

dttvtv

T

R


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The correlation process is illustrated below:

As previously stated:

VOUTv1(t)1

0T

V dtx

T

VX

VariableDelayv2(t) v2(t - )

R12()

2

2

21 )()(1

12

T

T

dttvtvT

R


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The output R12() is the correlation between the

two functions as a function of the delay .

The correlation at a particular value of would

be solved by solving R12(),


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AUTO CORRELATION AND CROSS CORRELATIONFUNCTIONS

Auto Correlation

In auto correlation a signal is compared to a time

delayed version of itself. This results in the Auto

Correlation Function or ACF.

Consider the function v(t), (which in general maybe random or deterministic).

The ACF, R() , is given by

2

2

)()(1

T

T

dttvtvT

R


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Of particular interest is the ACF when = 0, and

v(t) represents a voltage signal:

R(0) represents the mean square value or

normalised average power in the signal v(t)

2

2

2)(1

0

T

T

dttvT

R


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Cross Correlation

In cross correlation, two separate signals are

compared, eg the functions v1(t) and v2(t)previously discussed.

The CCF is

2

2

21 )()(1

12

T

T

dttvtvT

R


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Diagrams for ACF and CCF

Auto Correlation Function, ACF

Note, if the input is v1(t) the output is R11() if the input is v2(t) the output is R22()

v(t)1

0T

V dtx

T

VX

Variable

Delayv(t - )

R()


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Cross Correlation Function, CCF

v1(t)1

0T

V dtx

T

VX

VariableDelay

v2(t) v2(t - )

R12()


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CORRELATION COEFFICIENT

The correlation coefficient, , is the normalised

correlation function.

For cross correlation (ie the comparison of twoseparate signals), the correlation coefficient is

given by:

Note that R11(0) and R22(0) are the mean squarevalues of the functions v1(t) and v2(t) respectively.

)0().0(

)(

2211

12

RR

R


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For auto correlation (ie the comparison of a signal

with a time delayed version of itself), the

correlation coefficient is given by:

For signals with a zero mean value, is in the

range1 +1

)0(

)(

)0().0(

)(

R

R

RR

R


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If = +1 then the are equal (Positive correlation).

If = 0, then there is no correlation, the signals

are considered to be orthogonal.

If = -1, then the signals are equal and opposite

(negative correlation)


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EXAMPLES OF CORRELATION

CONTINUOUS TIME FUNCTIONS

v1(t)1

0T

V dtx

T

v2(t) = cost

R12(0)


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The above may be used for the demodulation of

PSK/PRK (Phase Shift Keying / Phase Reversal

Keying) signals.

For PSK/PRK, the input signal is v1(t) = d(t)cost,

d(t) = +V for data 1s and d(t) = -V for data 0s.

The second function, v2(t) = cost, is the carrier

signal.

Analyse the above process to determine the

output R12(0) for the inputs given.


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CORRELATIONDISCRETE TIME SIGNALS


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CORRELATION OF DIGITAL SIGNALS

Searching for Synchronisation Pattern 01111110, in

Data Bit Stream

Sync Data Sync

X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15

. . 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 1 0

0 1 1 1 1 1 1 0

Data In

Clock

S0 S1 S2 S3 S4 S5 S6 S7

Sum Agreements A Compare

Threshold T bits

Sync found if

A T bits

correlation correlation

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