111 lecture 2 signals and systems (ii) principles of communications fall 2008 nctu ee tzu-hsien sang
TRANSCRIPT
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111
Lecture 2Signals and Systems (II)
Principles of Communications
Fall 2008
NCTU EE Tzu-Hsien Sang
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Outlines
• Signal Models & Classifications
• Signal Space & Orthogonal Basis
• Fourier Series &Transform
• Power Spectral Density & Correlation
• Signals & Linear Systems
• Sampling Theory
• DFT & FFT
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More on LTI Systems• A system is BIBO if output is bounded, given
any bounded input.
• A system is causal if: current output does not depend on future input; or current input does not contribute to the output in the past.
3
conditionDirichlet ofelement main
|)(||)(||})(max{|
|})()(max{||})(max{|
dhdhtx
dtxhty
0for ,0)(
)()()()()(0
tth
dtxhdtxhty
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• Paley-Wiener Condition:
• Remarks: (1) |H(f)| cannot grow too fast.
(2) |H(f)| cannot be exactly zero over a finite band of frequency.
• 2nd ver.:
4
.1
ln
,for 0)( and ,)( If
2
2
dff
H(f)
tthdffH
,1
ln and )( If
2
2
df
f
H(f)dffH
).for 0)(( causal is )(such that tthfHH
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Eigenfunctions of LTI Systems
• Another way of taking complicated things part.• Instead of trying to find a set of orthogonal basis
functions, let’s look for signals that will not be changed “fundamentally” when passing them through an LTI system.
• Why?• Consider the key words: analysis/synthesis.• Note: Eigen-analysis is not necessarily consistent
with orthogonal basis analysis.5
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• If , where is a constant, then is the eigenvalue for the eigenfunction g(t).
• Let
6
)()}({ tgtgΗ
numbercomplex arbitrary an : ,)( sAetx st
.)( where,)(
])([)()( )(
dAehtx
AedehdAehty
s
ststs
).(])([)(then
,2Let
22 txAedehty
tfjs
tfjfj
i
ii
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• (Cross)correlation functions related by LTI systems:
• Note: In proving them, we use:
)(|)(|)( .4
)()()( .3
)()()()( .2
)()()()()( .1
2
*
fSfHfS
fSfHfS
RhhR
dRhRhR
xy
xyx
xy
xxyx
)()}({ fHh )()}({ ** fHh
Filters!!!
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• Since almost any input x(t) can be represented by a linear combination of orthogonal sinusoidal basis functions , we only need to input to the system to characterize the system’s properties, and the eigenvalue
carries all the system information responding to . (Frequency response!!!)
• In communications, signal distortion is of primary concern in high-quality transmission of data. Hence, the transmission channel is the key investigation target.
ftje 2
ftjAe 2
)()( 2 fHdteth ftj
ftjAe 2
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• Three major types of distortion caused by a transmission channel:
1. Amplitude distortion: linear system but the amplitude response is not constant.
2. Phase (delay) distortion: linear system but the phase shift is not a linear function of frequency. (Q: What good is linear phase?)
3. Nonlinear distortion: nonlinear system
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• Example: Group Delay
).()( ,)(
2
1)( :Definition fHf
df
fdfTg
delays.different
have componentsfrequency different constant a
not is )(phaselinear not is system theIf
constant. a is which ,)(
2)( system,linear aFor
0
00
fT
tfT
ftHf
g
g
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.componentsfrequency output theand
input ebetween th relations phase relative theshowsIt
.2
)()( :delay Phase
f
ffTp
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• Example: Ideal general filters
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• Realizable filters approximating ideal filters
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The Uncertainty Principle• It can be argued that a narrow time signal has
a wide (frequency) bandwidth, and vice versa:constant)()( bandwidthduration
2
11
)0(
)0(2
)0()(|)(|)0(2 and
,|)()(|)(|)0(
proof), anot is (thisargument area-equal By the
0
TWTX
xW
xdffXdffXWX
fXdttxdttxTx f
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Sampling Theory
• You’ve probably heard of “signal processing.” But how to process a signal?
• For instance, the rectifier – max{x(t), 0}.• But, how to do Fourier transform of an arbitrary
signal x(t)?• Computers seem a good idea. But computers can only
work on numbers.• We need to “transform” the signal first into numbers.• Q: Tell discrete signals from digital signals.
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• Ideal sampling signal: impulse train (an analog signal) , T: the sampling period
• Analog (continuous-time) signal:
• Sampled (continuous-time) signal:
n
nTtts
)(tx)(tx
nss
ns
ns
nTtnTxnTttx
nTttxtstxtx
k kssss
kss
kffXfkfffXf
kffffXfSfXfX
][)(
Hopefully the math becomes easier in ideal case. The concept actually is harder.
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• Aliasing: If The replicas of X(f) overlap in frequency domain. That is, the higher frequency components of overlap with the lower frequency components of X(f-fs).
2 .sf W
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– Nyquist Sampling Theorem:
• Let x(t) be a bandlimited signal with X(f) = 0 for . (i.e., no components at frequencies greater than W.) Then x(t) is uniquely determined by its samples if .
Wf ||
,2,1,0),(][ nnTxnx s
WT
fs
s 21
Undersampling: 2sf W
Oversampling: 2sf W
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• In other words, oversampling preserves all the information that x(t) contains. It is possible to reconstruct x(t) purely by its samples.
• Ideal reconstruction filter (interpretation in frequency domain:
• In time domain:
WfBWeB
fHfH s
ftj ,)2
()( 020
)()(
)()(
00
20
0
ttxHfty
efXHffY
s
ftjs
0 0( ) ( ) ( ) 2 ( )sinc[2 ( )]s s s sn n
y t x nT h t nT BH x nT B t t nT
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• Two types of reconstruction errors
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DFT & FFT
• You can view DFT as a totally new definition for a totally different set of signals. Or you can try to connect it to the Fourier Transform.
1,,1,0 ,
1,,1,0 ,1
1
0
2
1
0
2
NkexX
NneXN
x
N
n
N
nkj
nk
N
k
N
nkj
kn
1
0
1
0
/22 )()(|)()(
|)()(
N
n
nkN
N
n
Nnkj
N
kj
ezj
WnhenheHkH
zHeH j
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• Fast Fourier Transform (FFT) is not a new transform, it is simply a fast way to compute DFT. So, don’t use FFT to denote the object that you want to compute; only use it to denote the tool that you use to compute it. (Gauss knew the method already!)
• Application example: Fast convolution via FFT:
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1
0
1
0
)()(1
)(1
)(N
k
nkN
N
k
nkN WkXkH
NWkY
Nny