1 signals and systems lecture 14 multiplication property communication system
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Signals and SystemsLecture 14
•Multiplication Property•Communication System
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Chapter 4 Fourier Transform
§4.5 The Multiplication property (modulation property)
(调制特性)
r t s t p t 1
2R j S j P j
0 F0 j ts t e S j
F0 0 0
1 1cos
2 2x t t X j X j
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Chapter 4 Fourier Transform
Example 4.23
Determine the Fourier transform of the signal
2
sin sin / 2t tx t
t
2/1 jX
2
1
2
3
2
3
2
1 0
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Chapter 4 Fourier Transform
例 某连续时间 LTI 系统的单位冲激响应为
若输入信号 ,
试求整个系统的输出。
sincos6
th t t
t
k
x t t k
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Chapter 4 Fourier Transform
1h t 2h t x t
cos 4 t cos 6 t
A B C
例 如图所示系统中, 1
sin 4
t
h t tt
2
sin,
d th t
dt t
若输入信号 sin 2
t
x tt,试分别 A、 B、 C各点信号
的频谱。
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Chapter 4 Fourier Transform
§4.5.1 Frequency-Selective Filtering with Variable Center Frequency
中心频率可调的频率选择性滤波器
tytj ce
txtj ce
tf
0
1
WW
jH tr
jX
0
jY
0 c
jH
WW
0 WW
jR
W2
c
0 W2
jF
c
jG
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Chapter 8 Communication Systems
Communication Channel信道
SourceSignals
Modulator 调制器
Modulated Signals
Be best suited for transmitting
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Chapter 8 Communication Systems
Modulation : The general process of embedding an information
-bearing signal into a second signal.
Extracting the information-bearing signal.Demodulation :
Modulation :
Amplitude Modulation (AM)
Frequency Modulation (FM)
Phase Modulation (PM)
ModulatingSignal
Carrier Signal
ModulatedSignal
tctxty tx
tc
Amplitude Modulation
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Modulating system model:
x(t)
c(t)
y(t)=x(t)c(t)
x(t) --- modulating signal
c(t) --- Carrier signal
Chapter 8 Communication Systems
§8.1 Complex Exponential and Sinusoidal Amplitude Modulation
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8.1.1 Amplitude Modulation with Complex Exponential Carrier
Exponential carrier:
For convenience, let c=0, so
)()( cctjetc tj cetc )(
Output signal(modulated signal):tj cetxty )()(
)()(
)(2)(
)(*)(2
1)(
c
c
jjXjY
jC
jCjXjY
(1) Modulation Theory
Chapter 8 Communication Systems
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Chapter 8 Communication Systems
12
ttjxttx
etxty
cc
tj c
sin)(cos)(
)()(
(1) Implementation
Chapter 8 Communication Systems
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8.1.2 Amplitude Modulation with Sinusoidal signal
For convenience, choose c=0, so
ttxtctxty ccos)()()()(
)(2
1)(
2
1)(
)()()(
)(*)(2
1)(
cc
cc
jjXjjXjY
jC
jCjXjY
Chapter 8 Communication Systems
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Chapter 8 Communication Systems
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Chapter 8 Communication Systems
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Chapter 8 Communication Systems
§8.1 Complex Exponential and Sinusoidal Amplitude Modulation
1. cj tc t e
2. cos cc t t
t
tx
t
ty Envelope
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8.2 Demodulation for Sinusoidal AM
8.2.1 Synchronous demodulation
(1) Demodulation process
Chapter 8 Communication Systems
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)2(4
1)2(
4
1)(
2
1)( cc jjXjjXjXjW
In frequency domain:
ttxtx
ttx
tctytw
c
c
2cos)(2
1)(
2
1
cos)(
)()()(2
In time domain:
Expected signal: )()( jWortx
Chapter 8 Communication Systems
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Chapter 8 Communication Systems
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(2) Synchronous problem
Chapter 8 Communication Systems
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Time domain:
)2cos()(2
1)cos()(
2
1
)cos()cos()(
)cos()()(
ccccc
cccc
cc
ttxtx
tttx
ttytw
The output of lowpass filter: )cos()( cctx
Ideal output: x(t) )( desirediscc
When , it is referred to as synchronous
demodulation.cc
Chapter 8 Communication Systems
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8.2.2 Asynchronous demodulation
Amplitude-modulated signal:
tAtxty ccos])([)(
Chapter 8 Communication Systems
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Chapter 8 Communication Systems
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Chapter 8 Communication Systems
1. Suppose the modulator and demodulator are not synchronized in phase.
① Complex Exponential Carrier
§8.2.1 Synchronous Demodulation
c cj te
tx ty CommunicationChannel
Transmitter Modulation
ReceiverSynchronous Demodulation
LPF tx tw
dj te
2. Suppose the modulator and demodulator are not synchronized in frequency.
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Chapter 8 Communication Systems
② Sinusoidal Carrier
cct cos
tx ty CommunicationChannel
Transmitter Modulation
ReceiverSynchronous Demodulation
LPF tx tw
cos dt
1. Suppose the modulator and demodulator are not synchronized in phase.
2. Suppose the modulator and demodulator are not synchronized in frequency.
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§8.3 Frequency-Division Multiplexing(频分复用)
Chapter 8 Communication Systems
t1cos tx1
t2cos tx2
ty1BPF
1
ty2BPF
2
tncos
txn tynBPF
ty
n
jX1
M 0 M
M 0 M
jX 2
M 0 M
jX n
22 11 0
jY
nn
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Chapter 8 Communication Systems
Quadrature Multiplexing(正交多路复用)
tccos tx1
tcsin
tx2
tr
tcsin
tccos
LPF
LPF
ty1
ty2
M
MCommunication
Channel tr tw1
tw2
txty 11
txty 22
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Chapter 8 Communication Systems
§8.4 Single-Sideband Sinusoidal Amplitude Modulation
(单边带)
M 0
jX
Mtccos
tx ty
c 0
jY
cM2
M2
Uppersideband
Uppersideband
Lower sideband
Double Sideband Modulation (DSB)
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Chapter 8 Communication Systems
Retain the upper sideband
c 0 cM
Uppersideband
Uppersideband
M
c 0 c
Lower sideband
MM
Retain the lower sideband
Single Sideband Modulation (SSB)
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Chapter 8 Communication Systems
M 0
jX
M M 0
jX p
M
j
j
Figure 8.21 SSB system using a 90o phase-shift network.
0 j
0 j-
jH
tx ty
tcsin
ty2
ty1
txp
90o phase-shift
cos ct
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Chapter 8 Communication Systems
cpcp jjXj
jjXj
jY 2
1
2
12
ttxty cp sin2
M 0
jX p
M
j
j
c 0
jY1
c
2/1
c cLower Sideband
jY
2/1
2/1 jY2
c c
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Summary
Chapter 4 Fourier Transform
Multiplication Property and Communication Systems
AM and DSB、 SSB System
Modulation and Demodulation
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Readlist
• Signals and Systems:– 5.1, 5.2
• Question: – DTFT
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Chapter 8 Communication Systems
Homework:
8.1 8.3 8.22 8.28