Correlation to Psd

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Correlation, Energy SpectralDensity and Power

Spectral DensityChapter 8

5/2/05 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 2

Introduction

• Relationships between signals can be just asimportant as characteristics of individualsignals

• The relationships between excitation and/orresponse signals in a system can indicate thenature of the system


Flow Velocity Measurement

The relative timing of the two signals, p(t) and T(t),and the distance, d, between the heater and thermometer together determine the flow velocity.

Heater

Thermometer


Correlograms

Two HighlyNegativelyCorrelatedDT Signals

Correlogram

Signals


Correlograms

TwoUncorrelatedCT Signals

Correlogram

Signals


Correlograms

Two PartiallyCorrelatedDT Signals

Correlogram

Signals

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Correlograms

These two CTsignals are not

stronglycorrelated but

would be if onewere shifted intime the right

amount

Correlogram

Signals


CorrelogramsDT Sinusoids With

a Time DelayCT Sinusoids With

a Time Delay

Correlogram Correlogram

Signals Signals


Correlograms

Two Non-Linearly Related

DT Signals

CorrelogramSignals


The Correlation FunctionPositively

Correlated DTSinusoids with Zero

Mean

Uncorrelated DTSinusoids with

Zero Mean

NegativelyCorrelated DT

Sinusoids with ZeroMean



Correlated RandomCT Signals with

Zero Mean

Uncorrelated RandomCT Signals with

Zero Mean

NegativelyCorrelated Random

CT Signalswith Zero Mean



Correlated CTSinusoids with Non-

zero Mean

Uncorrelated CTSinusoids withNon-zero Mean

NegativelyCorrelated CT

Sinusoids with Non-zero Mean

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Correlated RandomDT Signals withNon-zero Mean

Uncorrelated RandomDT Signals withNon-zero Mean

NegativelyCorrelated Random

DT Signalswith Non-zero Mean


Correlation of Energy SignalsThe correlation between two energy signals, x and y, isthe area under (for CT signals) or the sum of (for DTsignals) the product of x and y.

The correlation function between two energy signals, xand y, is the area under (CT) or the sum of (DT) theirproduct as a function of how much y is shifted relative tox.

x t( )y* t( )dt!"

"

#

x n[ ]y* n[ ]n=!"

"

#or

Rxy !( ) = x t( )y* t +!( )dt"#

#

$

Rxy m[ ] = x n[ ]y* n + m[ ]n=!"

"

#or


Correlation of Energy SignalsThe correlation function for two energy signals is very similarto the convolution of two energy signals.

Therefore it is possible to use convolution to find the correlation function.

It also follows that

x t( )!y t( ) = x t " #( )y #( )d#"$

$

%

x n[ ]!y n[ ] = x n "m[ ]y m[ ]m="#

#

$or

Rxy !( ) = x "!( )#y !( )

Rxy m[ ] = x !m[ ]"y m[ ]or

Rxy !( ) F

" # $ X*f( )Y f( )

Rxy m[ ]F

! " # X*F( )Y F( )or


Correlation of Power SignalsThe correlation function between two power signals, xand y, is the average value of their product as a functionof how much y is shifted relative to x.

If the two signals are both periodic and their fundamentalperiods have a finite least common period,

where T or N is any integer multiple of that least commonperiod.

or

Rxy !( ) = limT"#

1

Tx t( )y* t +!( )dt

T$

Rxy m[ ] = limN!"

1

Nx n[ ]y* n + m[ ]

n= N

#

Rxy !( ) =1

Tx t( )y t +!( )dt

T"

Rxy m[ ] =1

Nx n[ ]y n + m[ ]

n= N

!or


Correlation of Power Signals

Correlation of periodic signals is very similar to periodicconvolution

where it is understood that the period of the periodicconvolution is any integer multiple of the least commonperiod of the two fundamental periods of x and y.

Rxy !( ) =x "!( ) y !( )

T

Rxy m[ ] =x !m[ ] y m[ ]

Nor

Rxy !( ) =x "!( ) y !( )

T

Rxy !( ) =x "!( ) y !( )

T

Rxy !( ) FS" # $ X

*k[ ]Y k[ ]

Rxy m[ ]FS

! " # X*k[ ]Y k[ ]or


Correlation of Power Signals

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Correlation of Sinusoids

• The correlation function for two sinusoids ofdifferent frequencies is always zero. (pp.588-589)


AutocorrelationA very important special case of correlation isautocorrelation. Autocorrelation is the correlation of afunction with a shifted version of itself. For energy signals,

At a shift, τ or m, of zero,

which is the signal energy of the signal. For power signals,

which is the average signal power of the signal.

Rxx!( ) = x t( )x t +!( )dt

"#

#

$

Rxxm[ ] = x n[ ]x n + m[ ]

n=!"

"

#or

Rxx0( ) = x

2t( )dt

!"

"

#

Rxx0[ ] = x

2n[ ]

n=!"

"

#or

Rxx0( ) = lim

T!"

1

Tx2t( )dt

T# or

Rxx0[ ] = lim

N!"

1

Nx2n[ ]

n= N

#


Properties of AutocorrelationAutocorrelation is an even function.

Autocorrelation magnitude can never be larger than it is atzero shift.

If a signal is time shifted its autocorrelation does not change.

The autocorrelation of a sum of sinusoids of differentfrequencies is the sum of the autocorrelations of theindividual sinusoids.

Rxx!( ) = R

xx"!( )

Rxxm[ ] = R

xx!m[ ]or

Rxx0( ) ! R

xx"( )

Rxx0[ ] ! R

xxm[ ]or


Autocorrelation ExamplesThree differentrandom DTsignals and theirautocorrelations.Notice that, eventhough thesignals aredifferent, theirautocorrelationsare quite similar,all peakingsharply at a shiftof zero.


Autocorrelation ExamplesAutocorrelations for a cosine “burst” and a sine “burst”.Notice that they are almost (but not quite) identical.


Autocorrelation Examples

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Matched Filters

• A very useful technique for detecting thepresence of a signal of a certain shape in thepresence of noise is the matched filter.

• The matched filter uses correlation to detectthe signal so this filter is sometimes called acorrelation filter

• It is often used to detect 1’s and 0’s in abinary data stream


Matched FiltersIt has been shownthat the optimalfilter to detect anoisy signal is onewhose impulseresponse isproportional to thetime inverse of thesignal. Here aresome examples ofwaveshapesencoding 1’s and0’s and the impulseresponses ofmatched filters.


Matched FiltersNoiseless Bits Noisy Bits

Even in the presence ofa large additive noisesignal the matched filterindicates with a highresponse level thepresence of a 1 and witha low response level thepresence of a 0. Sincethe 1 and 0 are encodedas the negatives of eachother, one matched filteroptimally detects both.



Two familiar DTsignal shapes andtheir autocorrelations.



Three random power signals with different frequency contentand their autocorrelations.


Autocorrelation ExamplesAutocorrelation functions for a cosine and a sine. Notice that the autocorrelation functions are identical even thoughthe signals are different.

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Autocorrelation Examples• One way to simulate a random signal is with

a summation of sinusoids of differentfrequencies and random phases

• Since all the sinusoids have differentfrequencies the autocorrelation of the sum issimply the sum of the autocorrelations

• Also, since a time shift (phase shift) does notaffect the autocorrelation, when the phasesare randomized the signals change, but nottheir autocorrelations



Let a random signal be described by

Since all the sinusoids are at different frequencies,

where is the autocorrelation of .

x t( ) = Ak cos 2!f0kt +"k( )k=1

N

#

Rx!( ) = R

k!( )

k=1

N

"

Rk!( )

Ak cos 2!f0 kt +"k( )



Four DifferentRandom Signals

with IdenticalAutocorrelations













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Cross CorrelationCross correlation is really just “correlation” in the cases inwhich the two signals being compared are different. Thename is commonly used to distinguish it from autocorrelation.


Cross Correlation

A comparison of x and y with y shifted for maximumcorrelation.


Cross CorrelationBelow, x and z are highly positively correlated and x and y areuncorrelated. All three signals have the same average signalpower. The signal power of x+z is greater than the signal powerof x+y.


Correlation and the Fourier Series

Calculating Fourier series harmonic functions can be thoughtof as a process of correlation. Let

Then the trigonometric CTFS harmonic functions are

Also, let

then the complex CTFS harmonic function is

c t( ) = cos 2! kf0

( )t( ) and s t( ) = sin 2! kf0

( )t( )

Xck[ ] = 2R

xc0( ) , X

sk[ ] = 2R

xs0( )

z t( ) = e+ j 2! kf

0( )t

X k[ ] = Rxz0( )


Energy Spectral DensityThe total signal energy in an energy signal is

The quantity, , or , is called the energy spectraldensity (ESD) of the signal, x and is conventionally given thesymbol, Ψ. That is,

It can be shown that if x is a real-valued signal that the ESD iseven, non-negative and real.

Ex

= x t( )2

dt!"

"

# = X f( )2

df!"

"

#

Ex

= x n[ ]2

n=!"

"

# = X F( )2

dF1$or

X f( )2

X F( )2

!xf( ) = X f( )

2

!xF( ) = X F( )

2or


Energy Spectral DensityProbably the most important fact about ESD is therelationship between the ESD of the excitation of an LTIsystem and the ESD of the response of the system. It can beshown (pp. 606-607) that they are related by

!y f( ) = H f( )2!x f( ) = H f( )H*

f( )!x f( )

!y F( ) = H F( )2!x F( ) = H F( )H*

F( )!x F( )

or

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Energy Spectral Density


Energy Spectral Density

It can be shown (pp. 607-608) that, for an energy signal,ESD and autocorrelation form a Fourier transform pair.

Rxt( ) F

! " # $xf( )

Rxn[ ]

F

! " # $xF( )or


Power Spectral DensityPower spectral density (PSD) applies to power signals inthe same way that energy spectral density applies toenergy signals. The PSD of a signal x is conventionallyindicated by the notation, or . In an LTIsystem,

Also, for a power signal, PSD and autocorrelation form aFourier transform pair.

Gxf( )

GxF( )

Gy f( ) = H f( )2Gx f( ) = H f( )H*

f( )Gx f( )

Gy F( ) = H F( )2Gx F( ) = H F( )H*

F( )Gx F( )

or

R t( ) F

! " # G f( )[ ] or

R n[ ]F

! " # G F( )


PSD Concept


TypicalSignals in

PSDConcept

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Correlation to Psd