1 11 lecture 8 basic modulation techniques (iv) fall 2008 nctu ee tzu-hsien sang
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TRANSCRIPT
22
Outlines
• Linear Modulation
• Angle Modulation
• Interference
• Feedback Demodulators
• Analog Pulse Modulation
• Delta Modulation and PCM
• Multiplexing
2
3
FM (Single Tone Wide-band Modulation)
0
sin
sin sin
00
Let ( ) sin .
Since it is a periodic signal, so is the FM signal.
Use Fourier series to represent the signal.
( ) cos( sin ) Re{ }
1
2
c m
m m m
m
j t j tc C c m C
T j t jn t j tmn
t t
x t A t t A e e
C e e dt e eT
( sin )
sin
1( )
2
( )
( ) Re{ ( ) } ( )cos[( ) ]
( ) ( ) ( ) ( ) ( ).
mm
m
m m
c m
jn t
j nx xn
j t jn tn
n
j t jn tc C n C n c m
n n
c n m c n m cn n
dt
e dx J
e J e
x t A e J e A J n t
X f J n J n
4
22 2 2 2 2
22 2 2 2
Power of an FM signal.
Can you guess the answer without referring to equations?
1 1( ) [ ( ) cos( ) ] ( ) .
2 2
Or,
1 1( ) ( cos( sin )) cos 2( sin ) .
2 2 2
c C n c m C n Cn n
Cc C c m C c m C
x t A J w nw t A J A
Ax t A w t w t A w t w t A
Bandwidth
When is small (narrowband FM), ( ) 0, for 2.
When is large (wideband FM), find the significant coeffients that contain
most of the signal power.
For example, a 98% power bandwidth 2(
nJ n
1) for the single tone.
1 ( )max
Peak frequency deviation 2 In general, define ,
BW of ( )
then the bandwidth 2( 1) .
m
t
f
d tdt
Dm t W
D W
7
Demodulation of Angle-Modulated Signals
• Ideal frequency discriminator: a device that yields an output proportional to the frequency deviation of the input.
( ) cos( ( )).
1 ( )( ) .
2
r c c
D D
x t A t t
d ty t K
dt
8
• Approximation:
( ) ( )( ) ( )sin( ( )).
( )( ) output of envelope detector 2 ( ).
rC c c
D c C d
dx t d te t A t t
dt dtd t
y t A A f m tdt
To reduce the channel noise effect, an amplitude limiter and a BPF are placed before the differentiator --- a band-pass limiter.
10
Interference
• Why study interference? The short answer: we don’t have the ability to study noise yet. Let’s start with something simpler.
• Interference means unwanted signals present in the signal path.
• Recall the possible causes of interference.
• Deterministic interference can be analyzed without using stochastic tools.
11
Interference in Linear ModulationAgain, we don't have the ability to deal with general-case signals.
Instead, a simple sinusoid message is assumed.
: cos
: cos( )
The received signal:
( ) cos c
mm
c
i c i
r C c m
AMessage t
A
Interference A t
x t A t A
os cos cos( )m c i c it t A t
12
1
1. Coherent Detection (linear)
( ) cos cos
2. Envelope Detection (non-linear)
1 1( ) Re{ [ ]}
2 2cos cos cos cos( )
[ cos cos ] co
c m m
D m m i i
j t j t j tj tr C i m m
C c m m c i c i
C m m i i
y t A t A t
x t e A Ae A e A e
A t A t t A t
A A t A t
s sin sinc i i ct A t t
13
Case (i) If (typical case)
( ) [ cos cos ] cos
Envelope of ( ) cos cos
( ) cos cos (same as the coherent detector)
Case (ii) If
( ) cos(
C i
r C m m i i c
r C m m i i
D m m i i
C i
r C c i
A A
x t A A t A t t
x t A A t A t
y t A t A t
A A
x t A
) cos cos( ) cos( )
[cos( ) cos sin( ) sin ] cos( )
cos [cos( ) cos sin( ) sin ]
[ cos cos cos
i m m c i i i c i
C c i i c i i i c i
m m c i i c i i
i C i m m i
t A t t A w t
A t t t t A t
A t t t t t
A A t A t t
] cos( )
[ sin cos sin ] sin( )
[ cos cos cos ] cos( )
Envelope of ( ) cos cos cos
( ) cos cos cos
c i
C i m m i c i
i C i m m i c i
r i C i m m i
D C i m m i
t
A t A t t t
A A t A t t t
x t A A t A t t
y t A t A t t
14
• The threshold effect: when the interference is greater than a certain level, the message is totally lost. It’s a non-linear phenomenon.
15
Interference in Angle Modulation
We are doing wrose here. No signal is assumed.
Assuming an interference at .
( ) cos( ( )) cos( )
cos cos cos sin sin
[ cos ]cos [ sin
c i
r C c i c i
C c i i c i i c
c i i c i
x t A t t A t
A t A t t A t t
A A t t A
2 2
1
]sin
( ) cos[ ( )]
( ) ( cos ) ( sin )where .sin
( ) tan ( )cos
i c
c
C i i i i
i i
C i i
t t
R t t t
R t A A t A t
A tt
A A t
16
1
Case (i) If (typical case) and assume an ideal discriminator.
( ) cos
sin sin( ) tan ( )
For PM: ( ) sin
1For FM: ( ) ( sin ) cos
2
C i
C i i
i i i i
C C
iD D i
C
i iD D i D i
C C
A A
R t A A t
A t A tt
A A
Ay t K t
A
A Ady t K t K f
dt A A
In summary,
1AM: cos
PM: sin .
FM: cos
i
ii
C
iD i
C
iD i i
C
t
A ta A
AK t
A
AK f t
A
17
i
Case (ii) If (very difficult to analyze)
Make a rough analysis based on phasor diagram.
( ) cos cos( )
Re{[ ] }
Interference phase (t)= . Resultant phase (
i c
C i
r C c i c i
j t j tC i
A A
x t A t A t
A Ae e
t
t).