### Phasor Analysis of Bandpass Signals

```Phasor Analysis of Bandpass Signals
Background: Phasors for Monochromatic Signals
x  t   A cos  2  f o t  

1
Phasor Analysis of Bandpass Signals
Narrowband Signals:
Define:
x t 
Z
 f   2 u t  X  f 
2
Narrowband Signals: Continued
3
Phasor Equivalent for Narrowband
4
Transmitting Bandpass Signals
through Bandpass Systems
x  t  : b an d p ass sig n al w ith cen ter freq u en cy f o
h  t  : im p u lse resp o n se o f L T I system
- n arro w b an d
- cen tered o n freq u en cy f o
F in d y  t  :
5
Amplitude Modulation
ECE460
Spring, 2012
Analog Modulation Techniques
Modulation: The process by which some
characteristic of a carrier wave is
varied in accordance with an
information-bearing signal
Amplitude modulation
1.
Amplitude modulation (AM)
2.
Double sideband-suppressed carrier (DSB-SC)
3.
Single sideband (SSB)
4.
Vestigial sideband (VSB)
7
Amplitude Modulation (AM)
AM is formally defined as a process in which the
amplitude of the carrier wave c(t) is varied
about a mean value linearly with a message
signal m(t).
Message Signal:
m (t )
Sinusoidal Carrier:
c ( t )  Ac co s(2  f c t   )
AM Wave:
s ( t )  Ac 1  k a m ( t )  cos(2  f c t   )
8
Amplitude Modulation
M essag e sig n al m ( t )
k a m (t )  1  t
k a m ( t )  1 fo r so m e t
9
Frequency Domain
Message signal: Band-limited to W
m (t )  M ( f )
AM wave:
Time Domain:
s ( t )  Ac 1  k a m ( t )  co s( 2  f c t   )
Frequency Domain:
S( f ) 
Ac
2
 ( f
 fc )   ( f  fc ) 
k a Ac
2
M ( f
 fc )  M ( f  fc )
10
11
Example
Message signal
m ( t )  A m co s( 2  f m t )
AM Wave
Time Domain:
s ( t )  Ac 1   cos(2  f m t )  cos(2  f c t )
 Ac cos(2  f c t ) 
1
2
 Ac cos  2 
 fc 
w here   k a Ac
f m  t  
1
2
 Ac cos  2 
 fc 
f m  t 
Frequency Domain:
S( f ) 
Ac
2

1
4

1
4
 ( f
 fc )   ( f  fc )
 Ac   ( f  f c  f m )   ( f  f c  f m ) 
 Ac   ( f  f c  f m )   ( f  f c  f m ) 
12
Varying m
S( f ) 
Ac
2

1
4

1
4
 ( f
 fc )   ( f  fc )
 Ac   ( f  f c  f m )   ( f  f c  f m ) 
 Ac   ( f  f c  f m )   ( f  f c  f m ) 
Ac  1
  0.5
f c  0 .4 H z
f m  0 .0 5 H z
  1.0
  2.0
13
Envelope Detection
14
Conclusions on AM
• Power
• Channel Bandwidth
• Complexity
15
Double Sideband-Suppressed
Carrier Modulation (DSB-SC)
Message signal:
Carrier Wave:
Transmit Signal:
m (t )
c (t )
s (t )  c (t ) m (t )
 Ac co s(2  f c t   ) m ( t )
S( f ) 
16
Why Coherent Detection?
s (t )
Product
modulator
v (t )
Low-pass
filter
vo (t )
Ac co s( 2  f c t   )
Local
Oscillator
v ( t )  Ac co s  2  f c t    s  t 
 Ac Ac co s  2  f c t  co s  2  f c t    m  t 

1
2
Ac Ac co s  4  f c t    m  t  
1
2
Ac Ac co s    m  t 
17
1
Product
modulator
Low-pass
filter
2
Ac co s    m  t 
co s( 2  f c t   )
Voltage Controlled
Oscillator
Phase
discriminator
-90
Phase Shifter
sin ( 2  f c t   )
Product
modulator
Low-pass
filter
1
2
Ac sin    m  t 
18
Conclusions on DSB-SC
• Power
• Channel Bandwidth
• Complexity
19
Single Modulation (SSB)
1. Creating an SSB signal via a Hilbert Transform
2. Filtering a DSB –SC signal
20
Filtering DSB-SC signal for SSB
DSB-SC Signal:
s D S B ( t )  Ac m ( t ) co s(2  f c t )
Filter:
 1,
H(f)
 0,
f  fc
o th erw ise
21
Find SSB Transmission Signal
Message Signal:
Begin by finding
m ( t )  Ac co s(2  f m t ),
fm  fc
mˆ ( t )
22
Demodulation of an SSB signal
Requires a phase coherent demodulator like a
DSB-SC demodulation
r  t  co s  2  f c t   s  t  co s  2  f c t   

1
2
Ac m  t  co s    
1
2
Ac mˆ  t  sin     d o u b le freq u en cy term s
Low-pass filtering leaves:
r  t  co s  2  f c t  
1
2
Ac m  t  co s    
1
2
Ac mˆ  t  sin   
23
Conclusions on SSB
• Power
• Channel Bandwidth
• Complexity
24
Vestigial Sideband Modulation
Compromise between DSB-SC and SSB:
– Keeps a trace, or vestige, of the other sideband
– A portion of the other sideband is transmitted
Transmitted Signal:
s ( t )   Ac m  t  co s  2  f c t    h  t 
Ac
S( f ) 
2

 M
f  fc   M

f  f c   H
f
v  t   s  t  cos  2  f c t 
V
f

1
2
 S
Ac
4

Ac
4
V
f
Ac
4

f  fc   S

 M

 M
 f  M 
M
f  f c  
f  2 fc   M
 f   H 
 f   H 
f  2 f c   H
f  fc   H


f  fc 
f  fc 
f  f c  
25
Vestigial Sideband Filter
26
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