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Ronald M. Pascual, Ph.D.

Quantifies the degree of similarity between one set

of data (or sequence) and another

May be computed through sum of products

Theory: Given two independent and random data

sequences, the sum of products will tend towards a

vanishingly small random number as the number of pairs

of points is increased.

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To make the result independent of the number of points

taken, normalization may be added as follows:

where: N = number of pairs of points taken

Example: Compute the correlation r12

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Cross correlation (correlation of twodifferent sequences) Pattern recognition

RADAR / SONAR signal processing (delayestimation)

uto correlation (correlation of a sequenceto itself) Test for randomness of a signal

Detection/Recovery of periodic signals buried innoise

Pitch estimation / correction in music recordings

Out-of-phase 100%

correlated waveforms

with zero correlation at

lag zero.

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)0()()(

)0()()(

121212

121212

rN

jjrjr

N

r

j

jrjr

true

true

The effect of the end-

effect on the cross-

correlation r12(j)

Pairs of waveforms of different magnitudes but equal

cross-correlations

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211

0

1

0

2

2

2

1

1212

)()(1

)()(

N

n

N

n

nxnxN

jrj

Example: Compute the correlation coefficients

12(j) and 34(j)

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Cross-correlation of a signal with itself.

1

0

1111 )()(1

)(N

n

jnxnxN

jr

where: j = lag

N= number of sampling points

Symmetry Autocorrelation is an evenfunction.

1

0

1111 )()(1

)(N

n

jnxnxN

jr

1

0

1111 )()(1

)(N

n

nxjnxN

jr

)()( 1111 jrjr

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The autocorrelation function of a periodicwaveform, with period T, is itself a periodicwaveform of period T.

For a periodic waveform, x(t), of period T,

)()( nTtxtx2/

2/

11 )()(1

lim)(

T

T

Tdttxtx

T

r

2/

2/

11 )()(1

lim)(

T

TT

dtnTtxtxT

r

)()( 1111 nTrr

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The autocorrelation of a random signal willhave its peak value at zero lag and will reduceto random fluctuation of small magnitudeabout zero for lags greater than about unity.

(constitutes a test for random waveforms)

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Energy Spectral Density

)()]([ 11 fGrF E

where GE(f) is the energy spectral density of a waveform.

It can be also shown that

Er )0(11where E is the total energy of the waveform.

Let v(n) = s(n) + q(n)

= sum of a signal s(n) and noise q(n)

)]()([)]()([1 1

0

jnqjnsnqnsN

rN

n

vv

1

0

1

0

1

0

1

0

)()(1

)()(1

)()(1

)()(1 N

n

N

n

N

n

N

n

vv jnqnq

N

jnsnq

N

jnqns

N

jnsns

N

r

)()()()( jrjrjrjrr qqqssqssvv

0 if s(n) and q(n)

are uncorrelated

0 if q(n) is

random

Autocorrelation of a

noisy signal emphasize

signal properties by

reducing the noise

content.

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Let s(n) = signal of period T

q(n) = random noise

S(n) = s(n) + q(n)

(n-kT) =periodic impulse train of period T

,...2,1,0)],[()]()([1

)(1

0

kjkTnnqnsN

jrN

n

s

)]()(...)2()2()()()0()0([1

)0( NqNsTqTsTqTsqsNrs

)](...)2()()0()0([1

)0( NqTqTqqNsN

rs

)(1

)0()0(/

0

kTqN

srTN

k

s 0 as Ninfinity

,...2,1,0)],)[()]()([1)(

1

0

kkTjnnqnsN

jr

N

n

s

Similarly, for other values of (j),

results to cancellation of noise while yielding s(n). Thus,

,...2,1,0),1(),...,2(),1(),0()( jforNssssjrs

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A signal lost in a noisy waveform may beestimated by:

1. Autocorrelating the waveform to find periodof the signal.

2. Cross-correlating the waveform with aperiodic impulse train of the same period asthe signal.

DSP (by Proakis & Manolakis)

DSP: A Practical Approach (by Jervis andIfeachor)

DSP Lecture notes (Prof. Edwin Sybingco,DLSU)