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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