Signals and Systems
Lecture 14
•Multiplication Property
•Communication System
1
Chapter 4
Fourier Transform
§4.5 The Multiplication property (modulation property)
（调制特性）
1
R  j  
S  j   P  j 
2
r t   s t  p t 
F
s  t  e j0 t 
 S  j    0  
1
1
x  t  cos  0 t  X  j    0    X  j    0  
2
2
F
2
Chapter 4
Fourier Transform
Example 4.23
Determine the Fourier transform of the signal
x t  
sin t sin  t / 2 
 t2
1 / 2 X  j 
3 1 0 1

2
2
2
3 
2
3
Chapter 4
例
Fourier Transform
某连续时间LTI系统的单位冲激响应为
sin  t
ht  
cos 6 t
t
若输入信号 x  t  

  t  k 
，
k 
试求整个系统的输出。
4
Chapter 4
例
Fourier Transform
如图所示系统中，h1  t     t  
若输入信号 x  t  
d  sin  t 
sin 4 t
, h2  t   

dt

t


t
sin 2 t
，试分别A、B、C各点信号
t
的频谱。
x t 
h1  t 
cos  4 t 
A
B
h2  t 
C
cos  6 t 
5
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
e j c t
xt 
G j 
0
R j 
W 0 W 
r t 
1
W
0
W

f t 

H  j 
X  j 
2W
 c
H  j 
yt 
e  j c t
Y  j 

W 0 W c
F  j 
 c
2W
0

6
Chapter 8
Communication Systems
Communication Channel
信道
Source
Signals
Modulator
调制器
Modulated
Signals
Be best suited for transmitting
7
Chapter 8
Modulation :
Demodulation :
Communication Systems
The general process of embedding an information
-bearing signal into a second signal.
Extracting the information-bearing signal.
Amplitude Modulation (AM)
Frequency Modulation (FM)
Modulation :
Phase Modulation (PM)
Modulating
Signal xt 
yt   xt ct 
ct 
Modulated
Signal
Carrier
Signal
Amplitude Modulation
8
Chapter 8
Communication Systems
§8.1 Complex Exponential and Sinusoidal Amplitude Modulation
Modulating system model:
c(t)
x(t)
y(t)=x(t)c(t)
x(t) --- modulating signal
c(t) --- Carrier signal
9
Chapter 8
Communication Systems
8.1.1 Amplitude Modulation with Complex
Exponential Carrier
(1) Modulation Theory
Exponential carrier:
c(t )  e
For convenience, let c=0, so
Output signal(modulated signal):
j ( c t  c )
j c t
c(t )  e
j c t
y(t )  x(t )e
1
Y ( j ) 
X ( j  ) * C ( j )
2
C ( j )  2 (   c )
Y ( j )  X ( j  j c )
10
Chapter 8
Communication Systems
11
Chapter 8
Communication Systems
(1) Implementation
y (t )  x(t )e j ct
 x(t ) cos  c t  jx(t ) sin  c t
12
Chapter 8
Communication Systems
8.1.2 Amplitude Modulation with Sinusoidal signal
For convenience, choose c=0, so
y(t )  x(t )c(t )  x(t ) cos  ct
1
Y ( j ) 
X ( j ) * C ( j )
2
C ( j )   (   c )   (   c )
Y ( j ) 
1
1
X ( j  j  c )  X ( j   j  c )
2
2
13
Chapter 8
Communication Systems
14
Chapter 8
Communication Systems
15
Chapter 8
Communication Systems
§8.1 Complex Exponential and Sinusoidal Amplitude Modulation
1. c t   e jct
2. c t   cos ct
xt 
t
yt 
Envelope
t
16
Chapter 8
Communication Systems
8.2 Demodulation for Sinusoidal AM
8.2.1 Synchronous demodulation
(1) Demodulation process
17
Chapter 8
Communication Systems
In time domain:
w(t )  y (t )c(t )
 x(t ) cos 2  c t

1
1
x(t )  x(t ) cos 2 c t
2
2
In frequency domain:
1
1
1
W ( j )  X ( j )  X ( j  j 2 c )  X ( j  2 j c )
2
4
4
Expected signal:
x(t ) or W ( j )
18
Chapter 8
Communication Systems
19
Chapter 8
Communication Systems
(2) Synchronous problem
20
Chapter 8
Communication Systems
Time domain:
w(t )  y (t ) cos( c t  c )
 x(t ) cos( c t   c ) cos( c t  c )

1
1
x(t ) cos( c  c )  x(t ) cos( 2 c t   c  c )
2
2
The output of lowpass filter:
Ideal output: x(t)
x(t ) cos( c  c )
( c  c is desired )
When  c  
, itc is referred to as synchronous
demodulation.
21
Chapter 8
Communication Systems
8.2.2 Asynchronous demodulation
Amplitude-modulated signal:
y(t )  [ x(t )  A] cos  ct
22
Chapter 8
Communication Systems
23
Chapter 8
Communication Systems
§8.2.1 Synchronous Demodulation
① Complex Exponential Carrier
xt 
yt 
e
j ct c 
Transmitter
Modulation
Communication
Channel
wt 
e

xt 
LPF
 j d t 

Receiver
Synchronous Demodulation
1. Suppose the modulator and demodulator are not
synchronized in phase.
2. Suppose the modulator and demodulator are not
synchronized in frequency.
24
Chapter 8
Communication Systems
② Sinusoidal Carrier
xt 
yt 
cosct   c 
Transmitter
Modulation
Communication
Channel
wt 
xt 
LPF
cos d t   
Receiver
Synchronous Demodulation
1. Suppose the modulator and demodulator are not
synchronized in phase.
2. Suppose the modulator and demodulator are not
synchronized in frequency.
25
Chapter 8
Communication Systems
§8.3 Frequency-Division Multiplexing（频分复用）
x1 t 
x2 t 

xn t 
cos 1t 1
BPF
cos 2t 
2
BPF
y1 t 
y2 t 
yt 

Y  j 
 n
 2
 1
 M
0 M
 M
0 M 
X 2  j 

X n  j 
cos  nt  n
yn t 
BPF

X1  j 
0
1
 M
2
0 M 

n 
26
Chapter 8
Communication Systems
Quadrature Multiplexing（正交多路复用）
x1 t 
cos ct
cos ct
r t  Communication
Channel
x2 t 
sin  c t
r t 
w1 t 
M
y1 t 
LPF
w2 t  M
LPF
sin  c t
y2 t 
y1 t   x1 t 
y2 t   x2 t 
27
Chapter 8
Communication Systems
§8.4 Single-Sideband Sinusoidal Amplitude Modulation
（单边带）
xt 
X  j 
yt 
cos ct
 M
0
M

Y  j 
Upper
sideband
Upper
sideband
Lower
sideband
 c
2M
0
c
2M
Double Sideband Modulation (DSB)

28
Chapter 8
Communication Systems
Retain the upper sideband
Upper
sideband
Upper
sideband
M
 c
c
0

M
Retain the lower sideband
Lower
sideband
 c
M
0
M
c
Single Sideband Modulation (SSB)

29
Chapter 8
Communication Systems
Figure 8.21 SSB system using a 90o phase-shift network.
cos ct
y1 t 
xt 
-j  0
H  j   
 j   0
90o
phase-shift
x p t 
0
M
y2 t 
sin  c t
X  j 
 M
yt 
j

 M
0
X p  j 
M 
j
30
Chapter 8
Communication Systems
y2 t   x p t sin  ct
1
1
Y2  j  
X p  j  j c  
X p  j  j c 
2j
2j
1/ 2
 c
c
1/ 2
1/ 2
 c
Y2  j 
 M

0
M 
j
Y1  j 
0
j
X p  j 
c

Y  j 
 c
Lower Sideband
c

31
Chapter 4
Fourier Transform
Summary
Multiplication Property and Communication Systems
AM and DSB、SSB System
Modulation and Demodulation
32
Readlist
• Signals and Systems:
– 5.1, 5.2
• Question:
– DTFT
33
Chapter 8
Communication Systems
Homework:
8.1 8.3 8.22 8.28
34