1 chapter 2. signals and spectra this chapter reviews one of the two pre-requisites for...
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Chapter 2. Signals and Spectra This chapter reviews one of the two pre-requisites for
communications research. Signals and SystemsProbability, Random Variables, and Random Processes
We use linear, particularly LTI, systems to develop the theory for communications.
Outline2.1 Line Spectra and Fourier Series2.2 Fourier Transform and Continuous Spectra2.3 Time and Frequency Relations2.4 Convolution2.5 Impulses and Transforms in the Limit2.6 Discrete Time Signals and the Discrete Fourier Transform
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o Communication Engineering 통신공학Step 1. Given a communication medium, we first analyze the channel and build a
mathematical model. 주어진 통신 매체에 따라 Channel 을 분석하고 모형을 만든다 .
Step 2. Using the model, we design the pair of a transmitter and a receiver that best exploits the channel characteristic. Channel 에 가장 효과적 신호처리를 할 수 있도록 Transmitter 와 Receiver 를 설계한다 .
ex) Modulation ( 변조 ) 과 Demodulation ( 복조 )
Encoding 과 Decoding
Multiplexing 과 Demultiplexing
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o Mathematical Tool for Signal Processing: Fourier Analysis
time domain frequency domain
analysis, synthesis, design
2.1 Line Spectra and Fourier Series
o Linear Time-Invariant system
)(tv )(th )(tg
)(
})({)(Then
)(Let
)()()()()(
jwt
jwtjw
jwt
ejwH
edehtg
etv
dtvhtvthtg
4
< 정현파 신호 (Sinusoidal Signal ) 의 표현 >
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:
sec]/[/:
:
)cos()(
o
oo
o
wT
f
phase
radfrequencyangularradianw
amplitudeA
t
twAtv
5
대한민국 1 호 라디오 ( 금성 A-501) 1959 년 , 금성사 김해수가 설계와 생산을 담당 . – 대한민국 역사 박물관
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Line spectrum of periodic signals
복소지수 (Complex exponential) 에 의한 sinusoidal wave 정현파 신호의 표현
복소수 ? Euler’s theorem/identity
of
A
phase
of
f
f
Amplitude
θjθe θj sincos
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따라서
tjwjtjwj
o
oo eeA
eeA
t(wAv(t)
22
)cos
2A
2A
of of
f
of of
f
8
]Re[)cos()( tjwj
ooeAetwAtv
Phasor 를 이용한 정현파 신호의 표현
Phasor representation is useful when sinusoidal signal is processed by real-in real-out LTI systems.
)( two
scfo
실수축
)cos( twA o
)cos( twA o
허수축
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Q1 왜 frequency domain 표현이 중요한가 ?
( 여러 가지 정현파형이 선형적으로 결합된 신호 )
tttw 70sin2-)4020cos(35)(
)(tw
t
)90352cos(2)40102cos(302cos5 ttt
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A1 Line Spectrum
“ 왜 Phase 는 Amplitude 보다 덜 중요한가 ? (phase time delay )
“ 모든 주기적 신호는 정현파 신호의 선형적 결합으로 표현될 수 있다 .”
Frequency content
Amplitude Phase
5
3 2
0 10 35 0 10 35
40
90
d
O
tN
f f
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o Periodic Signals ( 주기 신호 )
frequencylFundamentaT
fperiod주기T
정수m
ttvmTtv
O
oO
O
:1
);(:
:
),()(
Rectangular pulse trainFigure 2.1-7
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o Fourier Series어떠한 periodic signal 정현파 신호의 선형적 집합
Where
Phasor 표현
tnfj
nn
oectv 2)(
dtetvT
cO
o
T
tnfj
O
n 2)(1
o
cj
nn
nfat
ecc nargtwo-sided line spectrum
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주기함수의 주파수 특성 (Spectrum of periodic signals)
1. harmonics of fundamental frequency .
2.
3. 실함수 는
of
componentDCtvdttvT
cOT
O
o :)()(1
)(tv
oddnfcnfc
evennfcnfc
oo
oo
)(arg)(arg
)()(
of
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Spectrum of rectangular pulse train
with ƒ0 = 1/4 (a) Amplitude (b) Phase
Figure 2.1-8
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trigonometric Fourier series for real signals
매우 중요한 함수xx
x
ctnfcctv non
no
sin)c(sin
)arg2cos(2)(1
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Fourier-series reconstruction of a rectangular pulse train Figure 2.1-9
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Fourier-series reconstruction of a rectangular pulse train
Figure 2.1-9c
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Gibbs phenomenon at a step discontinuity
Figure 2.1-10
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Average Power of Periodic Signal
signalcomplexAns왜
dttvT
tv OTt
tO
?
)(1
)(2
2 1
1
) 1 (ion normalizat R
2
)cos()()2A
P
twAtv예 o
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Parseval’s Power Theorem
2
2
2
2
])(1
[
][)(1
)()(1
)(1
0
nnn
nn
nn
tnfj
To
tnfj
nnT
O
To
TO
ccc
cdtetvT
dtectvT
dttvtvT
dttvT
P
o
o
O
O
O
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2.2 Fourier Transforms and Continuous Spectra Fourier Transform 비주기 신호 or Energy signal
called the analysis equation.
)(tv
tdttvEnergy
2
)(
dtetvtvFfV ftj 2)()]([)(
Definition
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Inverse Fourier Transform
called the synthesis equation.
)(arg)(arg,)()(,)(.3
)()(.2
)()(.1
!)()(
)()]([)(
0
)(arg
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fVfVfVfVrealtvIf
dttvfV
efVfV
uniquefVtv
dfefVfVFtv
f
fVj
ftj
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Ex1 Rectangular pulse21
0{)(
t
othewiset
fA
f
fA
ff
Afj
fj
A
eefj
AdtAe
dtetAfV
fjfjftj
ftj
sinc
sin
sinsin22
}{2
)()(
)2
(2)2
(22
2
2
2
)(tv
2
2
A
1 f
)()( tAtv
1
t
)( fv
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Rectangular pulse spectrum V(ƒ) = A sinc ƒ
Figure 2.2-2
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Rayleigh’s Energy Theorem
Generally
Also called Parseval’s relation/theorem.
dffV
dttvE
2
2
)(
)(
dffWfV
dffWdtetv
dtdfefWtvdttwtv
ftj
ftj
)()(
)(])([
})(){()()(
2
2
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Duality Theorem
)()]([
)()(
fxtXFThen
fXtxLet F
WtAtX 2sinc)()예
W2
1
W2
1
W
A
2
WW ft
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2.3 Time and Frequency Relations Superposition Property
Time Delay
Time Scale Change
)]([)]([)]()([ tybFtxaFtbytaxF
useful tool for linear systems
dftjd efVttvF 2)()]([
0)(1
)]([
fVtvF
Slow PlaybackFast Playback
Low ToneHigh Tone
linear phase
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Frequency Translation/Shift and Modulation
)(])([ 2
c
tfj ffVetvF c
)(tv
)( fV
f
)( cffV
cf f
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continued
)(2
)(2
)]2cos()([ c
j
c
j
c ffVe
ffVe
tftvF
(a) RF pulse (b) Amplitude spectrumFigure 2.3-3
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Differentiation and Integration
In general
Example. Triangular pulse
)(2)]([ fVfjtvdt
dF
)(2
1])([
)()2()]([
fVfj
dvF
fVfjtvdt
dF
t
n
n
n
Principle of FM demodulator
differentiator
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2.4 Convolution Convolution Integral
Graphical interpretation of convolutionFigure 2.4-1
dtwvtwtv
)()()()(
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Result of the convolution in Fig. 2.4-1Figure 2.4-2
In general, convolution is a complicated operation in the TD.
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Convolution Theorems
)()()()(
)()()()(
)()()(
)()(
)()()()(
fWfVtwtv
fWfVtwtv
zvwvzwv
zwvzwv
vwwv
dtwvtwtv
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2.5 Impulses and Transforms in the Limit Dirac delta function
Thus
otherwiseo
totovdtttvt
t 21
2
1
)()()(
otot
or
dttdtt
)(
1)()(
35
Two functions that become impulses as 0
Figure 2.5-2
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Properties
)( dttA A
o tdt
)()()(.1 dd ttvtttv
)()()(.2 dd tvdttttv
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실제적 함수 (Practical Impulses)
tt
or
tt
sinc1
)(
1)(
)()()(lim
b
)()(lim
0
0
ovdtttv
ecause
tt
t2
2
1
t
1
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Fourier Transform of Power Signals
)(2
)(2
)cos(
)(][
)(sinclim)]([)(
sinc][
lim)(
)(
c
j
cj
c
ctjw
ffAe
ffeA
twA
ffAAeF
fAfAtvFfV
fAt
AF
tAtv
AtvDC
c
infinite energy
jeA
2je
A
2
cf cf fo
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From Fourier Series , Other periodic signals
n
ontnfj
nn nffcfVectv o )()()( 2
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2.6 Discrete Time Signals and Discrete Fourier Transform DT signal
DT periodic signal and DFTSAnalysis equationSynthesis equation
DFT, IDFTPeriodic extension and Fourier Series
DTFTAnalysis equationSynthesis equation
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Convolution using the DFTQ. We are given a convolution sum of two finite-length DT
signals. Each signal has support N_1, N_2. Find the finite-length (at most N_1+N_2-1) output of the convolution using DFT.
A. Choose N>= N_1+N_2-1. Compute DFT(x) and DFT(h). Perform entry-by-entry multiplication. Apply the inverse DFT. Done.
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HW #1 (Due on Next Tuesday 9/22. Please turn in handwritten solutions.)
2.7 Questions3462.1-9, 13 2.2-7, 102.3-8, 142.4-8, 152.5-102.6-4, 6