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Communication SystemsLecture 3

Dong In KimSchool of Info/Comm Engineering

Sungkyunkwan University

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OutlineEnergy spectral density and autocorrelation

For energy signalsPower spectral density and autocorrelation

For power signalsLinear systems

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Energy Spectral Density and Autocorrelation

Recall that for energy signals2 2 21( ) ( ) ( )

2E x d X d X f dfτ τ ω ω

π

∞ ∞ ∞

−∞ −∞ −∞= = =∫ ∫ ∫

energy spectral density (Joules/Hz)2( ) ( )G f X f

What is the time-domain counterpart for G(f)?[ ] [ ]1 1 * 1 1 *( ) ( ) ( ) ( ) ( )G f X f X f X f X f− − − − = = ∗ F F F F

* *( ) ( )X f x τ↔ − ( )Note: ( ) ( )X f x τ− ↔ −

2( ) ( ) j fX f x e dπ ττ τ∞

−

−∞= ∫

* * 2 * 2 '( ) ( ) ( ') 'j f j fX f x e d x e dπ τ π ττ τ τ τ∞ ∞

−

−∞ −∞= = −∫ ∫

Change of variable 'τ τ= −From

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[ ]1 *( ) ( ) ( )G f x xτ τ− = ∗ −F

To find , define *( ) ( )x xτ τ∗ − *( ) ( ),y xτ τ= −*

* *

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x x x y x t y t dt

x t x t dt x t x t dt

τ τ τ τ τ

τ τ

∞

−∞

∞ ∞

−∞ −∞

∗ − = ∗ = −

= − = +

∫

∫ ∫

Energy Spectral Density and Autocorrelation

Time-domain autocorrelation function:* * *( ) ( ) ( ) ( ) ( ) ( ) ( )x x x t x t dt x t x t dtφ τ τ τ τ τ

∞ ∞

−∞ −∞∗ − = − = +∫ ∫

2( ) ( ) ( )G f X f φ τ= ↔ Fourier transform pair

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Energy Spectral Density and Autocorrelation

If x(t) is real:

Autocorrelation measures the similarity between the signal and a delayed version of the signal.

* * *( ) ( ) ( ) ( ) ( ) ( ) ( )x x x t x t dt x t x t dtφ τ τ τ τ τ∞ ∞

−∞ −∞∗ − = − = +∫ ∫

( ) ( ) ( ) ( ) ( ) ( ) ( )x x x t x t dt x t x t dtφ τ τ τ τ τ∞ ∞

−∞ −∞∗ − = − = +∫ ∫

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Energy Spectral Density and Autocorrelation

Get energy E from

2( ) ( ) j fG f e dfπ τφ τ∞

−∞= ∫

2( ) ( ) ( )G f X f φ τ= ↔Relationship between G(f) and ( )φ τ

( )φ τ

2 2( ) ( ) ( )E x d X f df G f dfτ τ∞ ∞ ∞

−∞ −∞ −∞= = =∫ ∫ ∫

Parseval’s theorem:

(0) ( )G f df Eφ∞

−∞= =∫0τ = ⇒

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Example – a square pulse

( ) ?φ τ =

a

Ax(t)

2A a

τ

t

τ

Ax(t-τ)

tE=?

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Example – a gated sinusoid

Each period we get a peak (but smaller)

( ) ?φ τ =x(t)

τ

t

x(t-τ)

τt

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Example

~ 0 if and are completely unrelatedas in “noise”

What value of τ will give the biggest ?: every point lines up, all products positive

( )x t

*( ) ( ) ( )x t x t dtφ τ τ∞

−∞= −∫

( )x t ( )x t τ−

( )φ τ0τ =

t

( )x t τ−

τ

t

τ

( )φ τ

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ExampleFind the autocorrelation, energy spectral density and energy when ( ) ( )tx t e u tα−=

1( )Xj

ωα ω

=+

22 2

1 1 1( ) ( )G Xj j

ω ωα ω α ω α ω

= = ⋅ =+ − +

[ ]1( ) ( )x G fφ τ −=F from tables1

2e α τ

α−=

?E =

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OutlineEnergy spectral density and autocorrelation

For energy signalsPower spectral density and autocorrelation

For power signalsLinear systems

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Power Spectral Density and autocorrelation

Time-average autocorrelation function of power signals:

* *

* *

1( ) ( ) ( ) lim ( ) ( )2

1 ( ) ( ) lim ( ) ( )2

T

T T

T

T T

R x t x t x t x t dtT

x t x t x t x t dtT

τ τ τ

τ τ

→∞ −

→∞ −

− = −

= + = +

∫

∫

Periodic signals: 0

*

0

1( ) ( ) ( )T

R x t x t dtT

τ τ+∫In Chap 5, we will define the statistical (ensemble) average autocorrelation function for random signals, and the notation for statistical average: ( )x t

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*1( ) lim ( ) ( )2

T

T TR x t x t dt

Tτ τ

→∞ −+∫

*1Note: (0) lim ( ) ( )2

T

T TR x t x t dt P

T→∞ −= =∫

Power Spectral Density and autocorrelation

Power spectral density (of power signals):

[ ]

[ ]

2

-1 2

( ) ( ) ( )

( ) ( ) ( )

j f

j f

S f R R e d

R S f S f e df

π τ

π τ

τ τ τ

τ

∞−

−∞

∞

−∞

=

= =

∫

∫

F

F

( )0 ( )P R S f df∞

−∞= = ∫ : so S(f) is the power spectral density.

(compare with energy signals)

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Properties of autocorrelation:21 . R (0 ) ( ) : P o w er.x t=

R (0) R ( ) fo r a ll .τ τ≥

Power Spectral Density and autocorrelation

* *1 1( ) lim ( ) ( ) lim ( ) ( )2 2

T T

T TT TR x t x t dt x t x t dt

T Tτ τ τ

→∞ →∞− −− = +∫ ∫

*2 . R ( ) ( )Rτ τ− =

If x (t) rea l, R ( ) ( ) ( ) even .R Rτ τ τ− = ⇒23. lim ( ) ( ) if x(t) does not have periodic component.R x t

ττ

→∞=

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OutlineEnergy spectral density and autocorrelation

For energy signalsPower spectral density and autocorrelation

For power signalsLinear systems

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Linear Systems

Linear systems:

x(t) y(t)H

[ ][ ] [ ]

1 1 2 2

1 1 2 2

( ) ( ) ( )

( ) ( )

y t a x t a x t

a x t a x t

= +

= +

H

H H

Time invariant systems:[ ] [ ]0 0( ) ( ) ( ) ( )y t x t y t t x t t= ⇒ − = −H H

Linear time invariant (LTI) systems:Both linear and time invariant

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Linear SystemsImpulse response of an LTI system

[ ]( ) ( )h t tδ= H

( ) ( ) ( )x t x t dλ δ λ λ∞

−∞= −∫

Any input can be written in terms of ( )tδ

In freq domain: ( ) ( ) ( )Y f H f X f=

x(t) y(t)h(t)

Send x(t) to an LTI system h(t):[ ]( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

y t x t x t d x h t d

x t h d

λ δ λ λ λ λ λ

λ λ λ

∞ ∞

−∞ −∞

∞

−∞

= = − = −

= −

∫ ∫

∫

H H

Only need to know h(t) to get y(t)

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Input-Output Relationship for Spectral Density

*( ) ( ) ( )yG f Y f Y f=* *( ) ( ) ( ) ( )X f H f X f H f=

2 2 2( ) ( ) ( ) ( )xH f X f H f G f= =

x(t) y(t)h(t)

Energy signals:

Output spectral density is a scaled version of input spectral density.

2( ) ( ) ( )y xG f H f G f=

The same is true for power signals, proved in Chap 5.2( ) ( ) ( )y xS f H f S f=

( ) ( ) ( )Y f H f X f=

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Input-Output Relationship for Autocorrelation Function

[ ]

2

21 1

21 1

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

y x

y y x

x

h x

G f H f G f

R G f H f G f

H f G f

R R

τ

τ τ

− −

− −

=

=ℑ =ℑ =ℑ ⊗ℑ

= ⊗

x(t) y(t)h(t)

Energy signals:

Convolution property still holds for the autocorrelation function.The same is true for power signals.

( ) ( ) ( )y h xR R Rτ τ τ= ⊗

( ) ( ) ( )y t h t x t= ⊗

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Example: Autocorrelation Function

0

0

00 0

0

0

/2* *

/20

2 2/2*

/20

22

1( ) ( ) ( ) ( ) ( )

1

mn

T

x T

m n mj j tT T Tn mT n m

njT

nn

R x t x t x t x t dtT

x x e e dtT

x e

π τ π

δ

π τ

τ τ τ−

−∞ ∞

− =−∞ =−∞

∞

=−∞

− = −

=

=

∫

∑ ∑∫

∑

Time-average autocorrelation function of a periodic signal:

2 2

0

( ) , x n x nn n

nS f x f P xT

δ∞ ∞

=−∞ =−∞

= − = ∑ ∑

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Example: Power Spectral Density

Power spectral density of a periodic signal:

22

0

22

0 0

( ) ( )y nn

nn

nS f H f x fT

n nx H fT T

δ

δ

∞

=−∞

∞

=−∞

= −

= −

∑

∑

22

0y n

n

nP x HT

∞

=−∞

= ∑

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