11

Lecture 2Signals and Systems (II)

Principles of CommunicationsFall 2008

NCTU EE Tzu-Hsien Sang

Outlines

• Signal Models & Classifications• Signal Space & Orthogonal Basis• Fourier Series &Transform• Power Spectral Density & Correlation• Signals & Linear Systems• Sampling Theory• DFT & FFT

2

Examples

• Symmetry Properties of x(t) and Its Fourier FunctionFor real periodic x(t), For real aperiodic x(t),

3

*nn XX

)()( * fXfX

• Fourier Transform of Singular Functions is not an energy signal (hence doesn’t satisfy Dirichlet condition).However, its FT can be obtained by formal definition.

• Example: The FT of ?

4

)(t

,1)( FTt ),(1 fFT

,)( 020

fjFT AettA ),( 00 ffAAe FTtfj

n

nTt )( 0

• Fourier Transform of Periodic Signals—Periodic signals are not energy signals (don’t satisfy Dirichlet’s conditions). But we are doing it anyway (at least formally)…

• Given a periodic signal • Example-1: • Example-2:

5

n

tjnneXtx 0)(

n

n nffXfX )()( 0

tf02cos

n

nTt )( 0(A pulse train! What good are they for?)

6Note: This table uses “” instead of “f”. But it doesn’t hurt the fundamental facts.

7

Transform Pairs (There is something nice to know in life…)

8

9

10

• Let FT of an aperiodic pulse signal p(t) be

• We can generate a periodic signal x(t) by duplicating p(t) at every interval Ts, then

• From convolution theorem,

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)()}({ fPtp

n

sn

s nTtptpnTttx )()(*])([)(

nsss

nss

ns

nffnfPffPnfff

fPnTtfX

)()()()(

)(]})({[)(

Taking inverse FT of the eq. on previous page.

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n

tnfjss

nsss

nsss

ns

senfPfnffnfPf

nffnfPfnTtptxfX

21

11

)(})({)(

})()({)()()}({

n

tnfjss

ns

senfPfnTtp 2)()(

Poisson sum formula

Power Spectral Density & Correlation• Why should we care about the “frequency

components” of a signal?• For energy signals:

• The time-averaged autocorrelation function• The squared magnitude of the FT represents

the “energy” distributed on the frequency axis.

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T

TTdxxdxxxx

fXfXfXfXfG

)()(lim)()()()(

)](*[)]([)](*)([)}({)( 1111

energy. signal)0( E

• For power signals:

• For periodic power signals:

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signalpower periodic ifsignalpower aperiodic if

,)()(1

,)()(21lim

)()()(

0

*

0

*

*

T

T

TT

dttxtxT

dttxtxT

txxR

dffSR )()0( )}({)( RfS

n

n nffXRfS )(||)}({)( 02

“Power spectral density function”

• The functions () and R() measure the similarity between the signal at time t and t+.

• G(f) and S(f) represents the signal energy or power per unit frequency at freq. f.

• , • R() is even for real x(t): • If x(t) does not contain a periodic component:

• If x(t) is periodic with period T0, then R() is periodic in with the same period.

• S(f) is non-negative.15

,)()()0( 2 RtxpowerR ).0()}(max{ RR

).()()()( * RtxtxR

.)()(lim 2

||txR

fRfS ,0)}({)(

• Cross-correlation of two power signals:

• Cross-correlation of two energy signals:

• Remarks:16

T

TT

xy

dttytxT

tytxtytxR

)()(21lim

)()()()()(

*

**

dttytxxy )()()( *

),()( * yxxy RR )()( * yxxy

Signals & Linear Systems

• The standard input/output black box model for linear systems. Q: Why does it work?

• Linear: Satisfies superposition principle

• Time-invariant: Delayed input produces an output with the same delay.

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)(tx )(tyΗ )}({)( txHty

)()()}({)}({)}()({)(

212211

2211

tytytxHtxHtxtxHty

)()}({ 00 ttyttxH

Describing LTI Systems with Impulse Responses

• Let h(t) be the impulse response:

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)}.({)( tHth

dtxtx )()()(

dtHx

dtxHtxHty

)}({)(

})()({)}({)(

)()()()}({

)(*)()()()}({)(

000 ttydtthxttxH

thtxdthxtxHty

If time-invariant,

19Note: This example is a linear, but not time-invariant system.

• The convolution form holds iff LTI.• Duality of signal x(t) & system h(t):

• The Convolution Theorem:

Key application: generally is easier than …

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dtxhdthxty )()()()()(

)()(})()({)()}({ fXfHdtxhfYty

)()( fXfH)()( thtx