11 Lecture 2 Signals and Systems (II) Principles of Communications Fall 2008 NCTU EE Tzu-Hsien Sang.
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Transcript of 11 Lecture 2 Signals and Systems (II) Principles of Communications Fall 2008 NCTU EE Tzu-Hsien Sang.
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,
11
)()}({ fPtp
n
sn
s nTtptpnTttx )()(*])([)(
nsss
nss
ns
nffnfPffPnfff
fPnTtfX
)()()()(
)(]})({[)(
Taking inverse FT of the eq. on previous page.
12
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.
13
T
TTdxxdxxxx
fXfXfXfXfG
)()(lim)()()()(
)](*[)]([)](*)([)}({)( 1111
energy. signal)0( E
• For power signals:
• For periodic power signals:
14
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.
17
)(tx )(tyΗ )}({)( txHty
)()()}({)}({)}()({)(
212211
2211
tytytxHtxHtxtxHty
)()}({ 00 ttyttxH
Describing LTI Systems with Impulse Responses
• Let h(t) be the impulse response:
18
)}.({)( 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 …
20
dtxhdthxty )()()()()(
)()(})()({)()}({ fXfHdtxhfYty
)()( fXfH)()( thtx