density density and power spectral correlation, energy spectral

Correlation, Energy SpectralDensity and Power Spectral

Density

5/10/04 M. J. Roberts - All Rights Reserved 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

TwoCompletelyNegativelyCorrelatedDT Signals

Correlogram

Signals


Correlograms

TwoUncorrelatedCT Signals

Correlogram

Signals


Correlograms

Two PartiallyCorrelatedDT Signals

Correlogram

Signals


Correlograms

These two CTsignals are not

stronglycorrelated but

would be if onewere shifted intime the right

amount

Correlogram

Signals


Correlograms

DT Sinusoids Witha Time Delay

CT Sinusoids Witha Time Delay

Correlogram Correlogram

Signals Signals


Correlograms

Two Non-Linearly Related

DT Signals

Correlogram

Signals


The Correlation FunctionPositively CorrelatedDT Sinusoids with

Zero Mean

Uncorrelated DTSinusoids with

Zero Mean

Negatively CorrelatedDT Sinusoids with

Zero Mean


The Correlation FunctionPositively CorrelatedRandom CT Signals

with Zero Mean

Uncorrelated RandomCT Signals with

Zero Mean

Negatively CorrelatedRandom CT Signals

with Zero Mean


The Correlation FunctionPositively Correlated

CT Sinusoids withNon-zero Mean

Uncorrelated CTSinusoids withNon-zero Mean

Negatively CorrelatedCT Sinusoids with

Non-zero Mean


The Correlation FunctionPositively CorrelatedRandom DT Signalswith Non-zero Mean

Uncorrelated RandomDT Signals withNon-zero Mean

Negatively CorrelatedRandom DT Signalswith Non-zero Mean


Correlation of Energy SignalsThe correlation between two energy signals, x and y, is thearea under (for CT signals) or the sum of (for DT signals)the product of x and y*.

The correlation function between two energy signals, x andy, is the area under (CT) or the sum of (DT) that productas a function of how much y is shifted relative to x.

x y*t t dt( ) ( )−∞

∞

∫ x y*n nn

[ ] [ ]=−∞

∞

∑or

R x yxy*τ τ( ) = ( ) +( )

−∞

∞

∫ t t dt R x yxy*m n n m

n

[ ] = [ ] +[ ]=−∞

∞

∑or

In the very common case in which x and y are both real,

R x yxy τ τ( ) = ( ) +( )−∞

∞

∫ t t dt R x yxy m n n mn

[ ] = [ ] +[ ]=−∞

∞

∑or


Correlation of Energy Signals

The correlation function for two real energy signals is very similarto the convolution of two real energy signals.

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

It also follows that

x y x yt t t d( )∗ ( ) = −( ) ( )−∞

∞

∫ τ τ τ x y x yn n n m mm

[ ] ∗ [ ] = −[ ] [ ]=−∞

∞

∑or

R x yxy τ τ τ( ) = −( )∗ ( ) R x yxy m m m[ ] = −[ ] ∗ [ ]or

R X Yxy*τ( )← → ( ) ( )F f f R X Yxy

*m F F[ ]← → ( ) ( )For


Correlation of Power SignalsThe correlation function between two power signals, x andy, is the average value of the product of x and y* as afunction of 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. For real periodic signals these become

orR lim x yxy*τ τ( ) = ( ) +( )

→∞ ∫T TTt t dt

1R lim x yxy

*mN

n n mN

n N

[ ] = [ ] +[ ]→∞

=∑1

R x yxy*τ τ( ) = ( ) +( )∫

1

Tt t dt

TR x yxy

*mN

n n mn N

[ ] = [ ] +[ ]=∑1

or

R x yxy τ τ( ) = ( ) +( )∫1

Tt t dt

TR x yxy m

Nn n m

n N

[ ] = [ ] +[ ]=∑1

or


Correlation of Power Signals

Correlation of real periodic signals is very similar toperiodic convolution

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.

Rx y

xy τ τ τ( ) = −( ) ( )T

Rx y

xy mm m

N[ ] = −[ ] [ ]

orRx y

xy τ τ τ( ) = −( ) ( )T

Rx y

xy τ τ τ( ) = −( ) ( )T

R X Yxy*τ( )← → [ ] [ ]FS k k R X Yxy

*m k k[ ]← → [ ] [ ]FSor


Correlation of Power Signals


Correlation of Sinusoids

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


AutocorrelationA very important special case of correlation is autocorrelation.Autocorrelation is the correlation of a function with a shiftedversion 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.

R x xxx*τ τ( ) = ( ) +( )

−∞

∞

∫ t t dt R x xxx*m n n m

n

[ ] = [ ] +[ ]=−∞

∞

∑or

R xxx 02( ) = ( )

−∞

∞

∫ t dt R xxx 02[ ] = [ ]

=−∞

∞

∑ nn

or

R lim xxx 01 2( ) = ( )

→∞ ∫T TTt dt or R lim xxx 0

1 2[ ] = [ ]→∞

=∑

Nn NN

n


Properties of AutocorrelationFor real signals, autocorrelation 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.

R Rxx xxτ τ( ) = −( ) R Rxx xxm m[ ] = −[ ]or

R Rxx xx0( ) ≥ ( )τ R Rxx xx0[ ] ≥ [ ]mor


Autocorrelation Examples

Three differentrandom DTsignals and theirautocorrelations.Notice that, eventhough the signalsare different, 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.


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 and 0’sand the impulseresponses ofmatched filters.


Matched FiltersNoiseless Bits Noisy Bits

Even in the presence of alarge additive noisesignal this 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.



• One way to simulate a random signal is witha 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 cost A f tk k kk

N

( ) = +( )=

∑ 2 01

π θ

R Rx τ τ( ) = ( )=∑ kk

N

1

R k τ( ) A f tk k kcos 2 0π θ+( )



Four DifferentRandom Signals

with IdenticalAutocorrelations


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 thought ofas a process of correlation. Let

Then the trigonometric CTFS harmonic functions are

Also, let

then the complex CTFS harmonic function is

c cos and s sint kf t t kf t( ) = ( )( ) ( ) = ( )( )2 20 0π π

X R , X Rxsc sk k[ ] = ( ) [ ] = ( )2 0 2 0xc

z t e j kf t( ) = + ( )2 0π

X Rxzk[ ] = ( )0


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.

E t dt f dfx x X= ( ) = ( )−∞

∞

−∞

∞

∫ ∫2 2E n F dF

nx x X= [ ] = ( )

=−∞

∞

∑ ∫2 2

1or

X f( )2 X F( )2

Ψx Xf f( ) = ( )2Ψx XF F( ) = ( )2or


Energy Spectral DensityProbably the most important fact about ESD is the relationshipbetween the ESD of the excitation of an LTI system and theESD of the response of the system. It can be shown (pp. 606-607) that they are related by

Ψ Ψ Ψy x*

xH H Hf f f f f f( ) = ( ) ( ) = ( ) ( ) ( )2

Ψ Ψ Ψy x*

xH H HF F F F F F( ) = ( ) ( ) = ( ) ( ) ( )2

or


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.

Rx xt f( )← → ( )F Ψ Rx xn F[ ]← → ( )F Ψor


Power Spectral DensityPower spectral density (PSD) applies to power signals inthe same way that energy spectral density applies to energysignals. The PSD of a signal x is conventionally indicatedby the notation, or . In an LTI system,

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

Gx f( ) Gx F( )

G H G H H Gy x*

xf f f f f f( ) = ( ) ( ) = ( ) ( ) ( )2

G H G H H Gy x*

xF F F F F F( ) = ( ) ( ) = ( ) ( ) ( )2

or

R Gt f( )← → ( )[ ]F or R Gn F[ ]← → ( )F


PSD Concept


TypicalSignals in

PSDConcept

density density and power spectral correlation, energy spectral

Documents