Transcript Periodic Signals (주기 신호) Rectangular pulse train

 Chapter 2. Signals and Spectra
 This chapter reviews one of the two pre-requisites for
communications research.
Signals and Systems
Probability, Random Variables, and Random Processes
 We use linear, particularly LTI, systems to develop the theory for
communications.
 Outline
2.1 Line Spectra and Fourier Series
2.2 Fourier Transform and Continuous Spectra
2.3 Time and Frequency Relations
2.4 Convolution
2.5 Impulses and Transforms in the Limit
2.6 Discrete Time Signals and the Discrete Fourier Transform
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o Communication Engineering 통신공학
Step 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 Receiver를 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
v(t )
h(t )
g (t )

g (t )  h(t )  v(t )   h( )v(t   )d
Let
Then
v(t )  e jwt

g (t )  { h( )e
 jw
 H ( jw)e jwt
3
d}e jwt
< 정현파 신호 (Sinusoidal Signal ) 의 표현 >
v (t)  A cos(w o t   )
 t  
A : amplitude
w  : radian /angular frequency [rad / sec]
 : phase
fo 
1
To

w
o
2
4
대한민국 1호 라디오 (금성 A-501)
1959년, 금성사 김해수가 설계와 생산을 담당. –대한민국 역사 박물관
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 Line spectrum
of periodic signals
Amplitude
A
fo
f
phase

fo
f
 복소지수 (Complex exponential)에 의한 sinusoidal wave정현파 신호
의 표현
복소수?
Euler’s theorem/identity
e jθ  cosθ  j sinθ
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따라서
v(t)  A cos(wot   )
A j jw t A  j  jw t
 e e  e e
2
2
o
A
fo

2
 fo
f
 fo
2

7
fo
f
A
o
 Phasor를 이용한 정현파 신호의 표현
v(t )  Acos(wot   )  Re[ Ae j e jw t ]
o
허수축
fo c
s
  (wot )
실수축
A cos( wot   )
A cos(w
o
t  )
Phasor representation is useful when sinusoidal signal is
processed by real-in real-out LTI systems.
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Q1 왜 frequency domain 표현이 중요한가?
w (t )
t
w(t )  5  3cos(20t  40 ) - 2 sin 70t
(여러 가지 정현파형이 선형적으로 결합된 신호)
 5 cos2 0t  3cos(210t  40 )  2 cos(2 35t  90 )
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A1 Line Spectrum
Phase
Amplitude
90
5
3
0
10
40
2
35
0
f
10
35
f
Frequency content

 td
“왜 Phase는 Amplitude보다 덜 중요한가? (phase
time delay )
NO
“모든 주기적 신호는 정현파 신호의 선형적 결합으로 표현될 수 있다.”
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o
Periodic Signals (주기 신호)
v(t  mTO )  v(t ),    t  
m :정수
1
TO :주기 ( period ); f o  : Fundamental frequency
TO
Rectangular pulse train
Figure 2.1-7
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o Fourier Series
어떠한 periodic signal
v(t ) 
정현파 신호의 선형적 집합


cn e
n  
j 2nfo t
Where
1
cn 
TO

TO
v(t )e
 j 2nfo t
dt
Phasor표현
cn  cn e j arg c
at
n
two-sided line spectrum
nfo
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주기함수의 주파수 특성 (Spectrum of periodic signals)
1. harmonics of fundamental frequency f o .
fo
2.
1
co 
TO
3. 실함수

TO
v(t )
v(t )dt   v(t )  : DC component
는
c (  nfo )  c ( nfo )
arg c(nfo )   arg c(nfo )
even
odd
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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

v (t)  c o   2c n cos( 2nfo t  argc n )
n 1
sinc(x ) 
sinx
x
매우 중요한 함수
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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
1
 v (t ) 
TO
2

t1 TO
t1
(R  1
normalization )
2
v(t ) dt
왜 ? Ans complex signal
예)
v(t )  A cos(wot   )
A2
P
2
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
Parseval’s Power Theorem
1
P
TO

v (t ) dt
1
To

v(t )v  (t ) dt

1

TO

TO
TO



v (t )[  cne  j 2nf t ]dt
o
TO
[
n  

2
n  
1
To

c c
n  
n

n

To

v(t )e  j 2nf t dt ]cn
0

c
n  
2
n
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 2.2 Fourier Transforms and Continuous Spectra
 Fourier Transform
비주기 신호 or Energy signal
v(t )

t
2
Energy  v(t ) dt
Definition

V ( f )  F [v(t )]   v(t )e  j 2ft dt
called the analysis equation.
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 Inverse Fourier Transform
1
v (t)  F [V (f )] 


V (f )e j2 ftdf

called the synthesis equation.
v (t)  V (f )
unique !
1 . V (f )  V (f )e j argV (f )
2 . V (f )f 0 
3 . If
v (t)



v (t)dt
real,
V (f )  V (f ),  argV (f )  argV (f )
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Ex1 Rectangular pulse
1
0
( t )  {

t 
2
othewise
v(t )
1

v(t )  A ( t )

2
2
t


V ( f )   A ( t )e  j 2ft dt



 j 2f ( )
 j 2f (  )
A
2
2
 2 Ae  j 2ft dt 
{e
e
}
2
 j 2f
A
A

 2 j sinf  sinf
 j 2f
f
sinf
 A
f
 A sinc f

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v( f )
A
1

f
Rectangular pulse spectrum V(ƒ) = A sinc ƒ
Figure 2.2-2
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 Rayleigh’s Energy Theorem

2
E   v (t ) dt

  V ( f ) df
2
Generally




 j 2ft
v
(
t
)
w
(
t
)
dt

v
(
t
){
W
(
f
)
e
df }dt

 



 [ v(t )e j 2ft dt]W  ( f )df

 V ( f )W  ( f )df
Also called Parseval’s relation/theorem.
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 Duality Theorem
Let
F
x(t ) 
X( f )
Then
F[ X (t )]  x( f )
예) X (t)  A sinc 2Wt
A
2W
1
 2W
1
2W
W
t
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W
f
 2.3 Time and Frequency Relations
 Superposition Property
F[ax(t )  by(t )]  aF [ x(t )]  bF [ y(t )]
useful tool for linear systems
 Time Delay
F [v(t  t d )]  V ( f )e  j 2ftd
linear phase
 Time Scale Change
F [v(t )] 
Slow Playback
Fast Playback
1

V(
f

)
 0
Low Tone
High Tone
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 Frequency Translation/Shift and Modulation
F[v(t )e j 2f t ]  V ( f  fc )
c
V ( f  fc )
V( f )
v(t )
fc
f
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f
continued
e j
e j
F [v(t )cos(2f c t  )]  V ( f  f c )  V ( f  f c )
2
2
(a) RF pulse (b) Amplitude spectrum
Figure 2.3-3
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 Differentiation and Integration
d
F [ v(t )]  j 2f V ( f )
dt
Principle of FM demodulator
differentiator
In general
dn
F [ n v(t )] ( j 2f ) n V ( f )
dt
t
1
F [  v( )d ]
V( f )
j 2f
Example. Triangular pulse
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 2.4 Convolution
 Convolution Integral
v (t) w (t) 



v ()w (t  )d 
Graphical interpretation of convolution
Figure 2.4-1
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Result of the convolution in Fig. 2.4-1
Figure 2.4-2
In general, convolution is a complicated operation in the TD.
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 Convolution Theorems
v (t) w (t) 



v ( )w (t   )d 
v w  w v
v  (w  z )  (v  w ) z
v  (w  z )  (v  w )  (v  z )
v (t) w (t)  V (f )W (f )
v (t)w (t)  V (f )W (f )
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 2.5 Impulses and Transforms in the Limit
 Dirac delta function

t2
t1
v(t ) (t )dt  v(o)
t1  o  t2
o
otherwise
Thus




  (t )dt    (t )dt  1
or
 (t )  o
to
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Two functions that become impulses as  
0
Figure 2.5-2
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A
A  (t  td )
o
Properties
1.
2.
v(t )   (t  td )  v(t  td )



v(t ) (t  td )dt  v(td )
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td
t
 실제적 함수 (Practical Impulses)
  (t ) 
1

1 t
 
  

or
  (t ) 
2

2
t
1

1
t
sinc



lim(t)   (t)
 0
b ecause

lim v (t)(t)dt  v (o )
 0  
37
t
 Fourier Transform of Power Signals
DC  v(t )  A
infinite energy
t
v(t )  lim A  
 
 
t
F [ A  ]  A sinc f
 
V ( f )  F [v(t )]  lim A sinc f  A ( f )
 
F [ Ae jwct ]  A ( f  f c )
A j
Ae  j
A cos( wc t   ) 
e  ( f  fc ) 
 ( f  fc )
2
2
A  j
e
2
 fc
A j
e
2
o
fc
f
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 From Fourier Series , Other periodic signals
v(t ) 

c e
n  
n
j 2nfo t
 V( f ) 

 c  ( f  nf )
n  
n
o
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 2.6 Discrete Time Signals and Discrete Fourier Transform
 DT signal
 DT periodic signal and DFTS
Analysis equation
Synthesis equation
 DFT, IDFT
Periodic extension and Fourier Series
 DTFT
Analysis equation
Synthesis equation
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 Convolution using the DFT
Q. We are given a convolution sum of two finite-length DT
signals. Each signal has support N_1, N_2. Find the finitelength (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 Questions
3
4
6
2.1-9, 13
2.2-7, 10
2.3-8, 14
2.4-8, 15
2.5-10
2.6-4, 6
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