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School of Information Science and Engineering, Shandong University
Principles of the Communications
Chapter 5Chapter 5
Effect of Noise on AnalogEffect of Noise on Analog
Communication SystemsCommunication Systems
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Copyrights Zhu, Weihong
School of Information Science and Engineering, Shandong University
Principles of the Communications
Chapter 5 Contents
5.1 Effect of noise on linear-modulation systems
5.2 Carrier-phase estimation with a phase-locked
loop(PLL). 5.3 Effect of noise on angle modulation
5.4 Comparison of analog-modulation systems 5.5 Effects of transmission losses and noise in
analog communication systems
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Copyrights Zhu, Weihong
School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
System model
Fig5.1.1 Block diagram of the demodulator
BPFum(t)
n(t)
ni(t) no(t)LPF+
( )u t
cos(2 )c f t π φ +
( ) y t ( )o y t
( ) ( ) ( )m ir t u t n t = +
( ) ( )cos2 ( )sin 2i c c s cn t n t f t n t f t π π = −
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Copyrights Zhu, Weihong
School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
c f W +c f c f −c f W − −
BPF-USSB
f
c f c f − c f W +c f W −c f W − +c f W − −
BPF-DSB
f
Figure 5.1.2 BPF
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
Comparison standard.
SNR (signal-to-noise ratio) of the output of the receiver is
the base coefficient to evaluate the analog communication
system.
In order to compare the effect of noise on various types of
analog-modulated signals, we also consider the effect of
noise on an equivalent baseband communication system
or we can see the input SNR of the demodulator.
LPF+m(t)
n(t)
m(t)+n(t)
Baseband system
N 0 /2
W/2 -W/2
Figure 5.1.3 Baseband system
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
00
2o
W
n
W
N P df N W
−= =∫
The power of the noise is
If we denote the received power by P R, the baseband SNR
is given by
0
R
b
S P
N N W
⎛ ⎞ =⎜ ⎟⎝ ⎠
5.1.1 Effect of Noise on a Baseband System (or input SNR of
the demodulator
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
5.1.2 Effect of Noise on DSB-SC AM
( ) ( )cos(2 )m c c cu t A m t f t π φ = +
[ ]
[ ]
( ) ( ) cos(2 ) ( ) cos 2 ( )sin 2( ) ( ) cos(2 )
( ) cos(2 ) ( ) cos 2 ( )sin 2 cos(2 )
1 1( ) cos( ) ( ) cos(4 )
2 2
1( )cos ( )sin
21
2
c c c c c s c
c
c c c c c s c c
c c c c c
c s
r t A m t f t n t f t n t f t y t r t f t
A m t f t n t f t n t f t f t
A m t A m t f t
n t n t
π φ π π π φ
π φ π π π φ
φ φ π φ φ
φ φ
= + + −= +
= + + − +
= − + + +
+ +
+ [ ]
[ ]
( ) cos(4 ) ( )sin(4 )
1( ) ( ) ( )2
c c c s c c
o c c c
n t f t n t f t
y t A m t n t if
π φ φ π φ φ
φ φ
+ + − + +
= + =
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
For DSB-SC AM signal, the demodulator is coherent or
synchronous demodulator.21
4
104
2 2
0 0
0
2
12 2
o
c mo
oDSB n
c m c m
R
b
PPS
N P WN
A P A PWN N W
P S
N W N
⎛ ⎞= =⎜ ⎟
⎝ ⎠
= = ×
⎛ ⎞= = ⎜ ⎟
⎝ ⎠
2
0 0
1 1 1
4 4 4
1
4 22
o co c m n n n
n
P A P P P P
where P N W N W
= = =
= × =
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
In DSB-SC AM , the output SNR is the same as the SNR
for a baseband system. Therefore, DSB-SC AM does not
provide any SNR improvement over a simple baseband
communication system.
Why do we still use DSB-SC AM system?
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
5.1.3 Effect of Noise on SSB AM
ˆ( ) ( )cos2 ( )sin 2c c c cu t A m t f t A m t f t π π = ±
ˆ( ) ( ( ) ( )) cos 2 ( ( ) ( ))sin 2( ) ( ) cos(2 ) ( ) cos 2 ( )sin 2
( ) ( ) cos(2 )
ˆ( ) cos(2 ) ( )sin(2 )
( ) cos 2 ( ) sin 2
c c c c s c
c c c c c s c
c
c c c c c c
c c s c
r t A m t n t f t A m t n t f t r t A m t f t n t f t n t f t
y t r t f t
A m t f t A m t f t
n t f t n t f t
π π π φ π π
π φ
π φ π φ
π π
= + + ± −= + + −
= +
+ ± +⎡ ⎤= ⎢ ⎥+ −⎣ ⎦
[ ]
[ ]
[ ]
1 1 12 2 2
1 12 2
12
1
2
cos(2 )
ˆ( ) cos( ) ( )sin( ) ( ) cos(4 )
ˆ ( )sin(4 ) ( ) cos ( )sin
( ) cos(4 ) ( )sin(4 )
( ) ( ) ( )
c
c c c c c c c
c c c c s
c c c s c c
o c c c
f t
A m t A m t A m t f t
A m t f t n t n t
n t f t n t f t
y t A m t n t if
π φ
φ φ φ φ π φ φ
π φ φ φ φ
π φ φ π φ φ
φ φ
+
= − + − + + +
+ + + + +
+ + + − + +
= + =
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
The SNR in a SSB system is equivalent to that of a DSB systemThe SNR in a SSB system is equivalent to that of a DSB system
Actrually2
02
c m
oDSB
S A P N WN
⎛ ⎞ =⎜ ⎟⎝ ⎠
2
0
c m
oSSB
S A P N WN
⎛ ⎞ =⎜ ⎟⎝ ⎠
How to explain the upper conclusion?
0 0
2
0 0
1
22n
o c m R
oSSB bn
P N W N W
P A P PS S
N P N W N W N
= × =
⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
21 1 1
4 4 4o co c m n n nP A P P P P= = =
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
5.1.4 Effect of Noise on Conventional AM
( ) [1 ( )]cos2m c n cu t A am t f t π = +
( ) [ [1 ( )] ( )]cos 2 ( )sin 2c n c c s cr t A am t n t f t n t f t π π = + + −
If the demodulator is a coherent demodulator
1( ) { [1 ( )] ( )}2
o c n c y t A am t n t = + +
2 21
4 no c mP A a P= 02nP WN =
1 1
4 4o cn n nP P P= =
22[1 ]
2 n
c R m
AP a P= +
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
2 2 2 214
104
2
n nc m c m
oAM n
A a P A a PS
N P N W
⎛ ⎞= =⎜ ⎟
⎝ ⎠
222
2
2
0
1
1
c
nn
n
A
mm
m
a Pa P
a P N W
⎡ ⎤+⎣ ⎦=+
2
2
01
n
n
m R
m
a P P
a P N W =
+
b
S N
η ⎛ ⎞= ⎜ ⎟⎝ ⎠
2
21
n
n
m
m
a Pa P
η =+
where
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
ηis less than 1. So the SNR in conventional AM is
always smaller than the SNR in a baseband system.
For the envelope detector, we can only obtain
approximational results.
The output of the envelope detector can be written as
[ ]( ) { (1 ( ) ( )}cos 2 ( )sin 2c n c c s c
r t A am t n t f t n t f t π π = + + −
[ ]{ }2 2
( ) 1 ( ) ( ) ( )c n c sV t A am t n t n t = + + +
S h l f I f i S i d E i i Sh d U i i
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
1. High input SNR or high P R, , which means
[ ]{ }( ( )) 1 ( )s c n
P n t A am t +
So [ ]{ }( ) 1 ( ) ( )c n c
V t A am t n t ≈ + +
Here SNR is equal to the coherent demodulator2 2
02
nc m
oAM
A a PS
N N W
⎛ ⎞=⎜ ⎟
⎝ ⎠ b
S
N η
⎛ ⎞= ⎜ ⎟
⎝ ⎠
2. Low input SNR or low P R
[ ]{ }( ( )) 1 ( )c n
P n t A am t +
S h l f I f i S i d E i i Sh d U i i
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
[ ]{ }
( ) ( )
( ) ( )
2 2
22 2 2
2 2
2 2
( ) 1 ( ) ( ) ( )
1 ( ) ( ) ( ) 2 ( ) 1 ( )
2 ( )( ) ( ) 1 1 ( )
( ) ( )
c n c s
c n c s c c n
c cc s n
c s
V t A am t n t n t
A am t n t n t A n t am t
A n t n t n t am t
n t n t
= + + +
= + + + + +
⎡ ⎤≈ + + +⎢ ⎥
+⎣ ⎦
[ ]( )( ) 1 ( )( )
c cn n
n
A n t V t am t
V t ≈ + +
Here the signal component is multiplied by noise and isno longer distinguishable. In this case, no meaningful
SNR can be defined. It is said that this system is
operating below the threshold (门限
).
S h l f I f ti S i d E i i Sh d U i it
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
Example 5.1.1 The message signal m( t) has a bandwidth
of 10kHz, a power of 16W and a maximum amplitude of
6. It is desirable to transmit this message to a
destination via a channel with 80dB attenuation and
additive white noise with power-spectral density
Sn( f )=N0 /2=10-12W/Hz, and achieve a SNR at the
demodulator output of at least 50dB. What is therequired transmitter power and channel bandwidth if
the following modulation schemes are employed?
1. DSB AM2. SSB AM
3. Conventional AM with modulaiton index equal to 0.8
S h l f I f ti S i d E i i Sh d U i it
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
Solution. We first determine (S/N)b as a basis of
comparison.
Since the channel attenuation is 80dB, the ration of
transmitted power PT to received power PR is
1. For DSB AM, we have
( )8
12 4
00.5 102 10 10
R R R
b
P PS P N N W
−
⎛ ⎞= = = ×⎜ ⎟ × ×⎝ ⎠
810lg 80 102
T T R T
b R
P PS P P
P N
− ⎛ ⎞= ⇒ = ⇒ =⎜ ⎟
⎝ ⎠
550 10 2002
2 20
T T
O R
PS S dB P KW
N N
BW W KHz
⎛ ⎞ ⎛ ⎞= = = ⇒ =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= =
∼
S h l f I f ti S i d E i i Sh d U i it
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Copyrights Zhu, Weihong
School of Information Science and Engineering, Shandong University
Principles of the Communications
5.1 Effect of Noise on Linear-modulation Systems
2. For SSB AM,
3. For conventional AM, with a=0.8,
550 10 200
210
T T
O R
PS S dB P KW
N N BW W KHz
⎛ ⎞ ⎛ ⎞= = = ⇒ =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠= =
∼
( )( ) ( )
2 2
22 2
5
2
0.8 16 / 360.22 where1 1 0.8 16 / 36 max ( )
0.22 10 909 2 20
2
n
n
n
T
O b
m
mm
m
T T
O
PS S
N N
a P PPa P m t
PS P KW BW W KHz
N
η η
η
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= = ≈ =+ +
⎛ ⎞≈ = ⇒ ≈ = =⎜ ⎟
⎝ ⎠
School of Information Science and Engineering Shandong Uni ersit
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
5.2.1. Principle of Phase-locked loop (PLL)
f(t)
F(f)
VCO
r(t)
x(t)
e(t)
y(t)
Phase detector
Loop filter
Voltage-controlled oscillator
Figure 5.2.1 Schematic of the basic phase-locked loop
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
Phase-locked loops are servo-control(伺服控制)
loops,
whose controlled parameter is the phase of a locally
generated replica of the incoming carrier signal.
Phase detector is a device that produced a measure of the
difference in phase between an incoming signal and the
local replica.
Loop filter governs the PLL’s response to the variations
in the error signal.
VCO is the device that produced the carrier replica. It is
a sinusoidal oscillator whose frequency is controlled by a
voltage level at the device input.
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
A VCO is an oscillator whose output frequency is a linearfunction of its input voltage over some range of input andoutput. A positive input voltage will cause the VCO
output frequency to be greater than its uncontrolledvalue, f 0, while a negative voltage will cause it to be less.
Phase lock is achieved by feeding a filtered version of thephase difference between the incoming signal r(t) and theoutput of the VCO, x(t), back to the input of the VCO ,y(t).
Consider a normalized input signal of the form
[ ]0( ) cos 2 ( )r t f t t π θ = +
Where f 0 is the nominal carrier frequency andθ(t) is
a slowly varying phase.
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
Consider a normalized VCO output of the form
0ˆ( ) 2sin 2 ( ) x t f t t π θ ⎡ ⎤= − +⎣ ⎦
These signals will produce an output error signal at the
phase detector output of the form
[ ]0 0
0
ˆ( ) ( ) ( ) 2sin 2 ( ) cos 2 ( )
ˆ ˆsin ( ) ( ) sin 4 ( ) ( )
e t x t r t f t t f t t
t t f t t t
π θ π θ
θ θ π θ θ
⎡ ⎤= = − + +⎣ ⎦
⎡ ⎤ ⎡ ⎤= − − + +⎣ ⎦ ⎣ ⎦
After the loop filter
ˆ ˆ( ) sin( ( ) ( )) ( ) ( ) y t t t t t θ θ θ θ = − ≈ −
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
If we make the assumption that f o is the uncontrolled
frequency of the VCO, we can express the difference in
the VCO output frequency from f 0 as the time differential
of the phase term . The output frequency of the VCOis a linear function of the input voltage. Therefore, since
an input voltage of zero produces an output frequency of
f 0, the difference in the output frequency from f 0 will beproportional to the value of the input voltage y(t), or
ˆ( )t θ
0 0
1 ˆ( ) ( ) ( )
2ˆ( ) ( ) ( ) ( ) ( )
d f t t Ky t
dt
K e t f t K t t f t
θ
π
θ θ
⎡ ⎤Δ = =⎢ ⎥
⎣ ⎦⎡ ⎤= ∗ ≈ − ∗⎣ ⎦
Where K 0 /2п
is the gain of the VCO.
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
Consider the Fourier transform of the upper equation
( ) ( ) ( )0ˆ ˆ( ) j K F ωθ ω θ ω θ ω ω ⎡ ⎤= −⎣ ⎦
( )( )
( )( )
0
0
ˆ( )
K F H
j K F
θ ω ω ω
θ ω ω ω = =
+
Steady-state tracking characteristics (稳态跟踪特征
)
( )( ) E e t ω = ⎡ ⎤⎣ ⎦F
( ) ( )[ ] ( )
( )
( )0
ˆ
1 ( ) H
j
j K F
θ ω θ ω ω θ ω
ωθ ω
ω ω
= −
= −
= +
So the phase
error could
be expressed
as
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
Using the final value theorem of Fourier transforms,which is
0lim ( ) lim ( )t j
e t j E ω
ω ω →∞ →
=
( ) ( )( )
2
00
lim ( ) limt j
je t
j K F ω
ω θ ω
ω ω →∞ →=
+
We get
This equation provides a measure of a loop’s ability to
cope with various types of changes in the input.
☺Example1 Response to a Phase Step
Consider a loop’s steady-state response to a phase
step at the loop input.
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
SolutionAssuming that the PLL was originally in phase lock, a phase
step will throw the loop out of lock. Having abruptlychanged, however, the input phase again becomes stable.This should be the easiest type of phase disturbance for aPLL to deal with. The Fourier transform of a phase stepwill be taken to be
( ) ( ( ))u t j
φ θ ω φ
ω
Δ= Δ =F
Where is the magnitude of the step. So
( ) ( )( ) ( )
2
0 00 0
lim ( ) lim lim 0t j j
j je t j K F j K F ω ω
ω θ ω ω φ ω ω ω ω →∞ → → Δ= = =+ +
Assuming the F(0)≠0. Thus the loop will eventually track
out any phase step that appears at the input if the loop hasa nonzero dc response.
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School of Information Science and Engineering, Shandong University
Principles of the Communications
5.2 Carrier-phase estimation with a phase-locked loop
☺Example 2. Response to a Frequency Step
Next, consider a loop’s steady-state response to a frequencystep at the input.
Solution
Since phase is the integral of frequency, the input phase willchange linearly as a function of time for a constant input-
frequency offset. The Fourier transform of the phasecharacteristic will be the transform of the integral of thefrequency characteristic. Since the frequencycharacteristic is a step, and the transform of an integral is
the transform of the integrand divided by the parameter jω
, it follows that( )
2( ) j
ω θ ω
ω
Δ=
Whereω
is the magnitude of the frequency step. So
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5.2 Carrier-phase estimation with a phase-locked loop
The steady-state result in this case depends on more
properties of the loop filter than merely a nonzero dc
response.. If the filter is “all-pass”, then F( ω
)=1
If it is a low-pass, then
( ) ( )( ) ( ) ( )
2
0 00 0 0
lim ( ) lim lim0t j j
je t
j K F j K F K F ω ω
ω θ ω ω ω
ω ω ω ω →∞ → →
Δ Δ= = =+ +
( ) 1
1F j
ω
ω ω ω = +
If it is a lead –lag(超前滞后
) filter, then ( ) 1 2
2 1
jF
j
ω ω ω ω
ω ω ω
⎛ ⎞ += ⎜ ⎟
+⎝ ⎠so
0
lim ( )t
e t K
ω
→∞
Δ=
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5.2 Carrier-phase estimation with a phase-locked loop
This steady-state error will exist regardless of the
order of the filter, unless the denominator of
F( ω ), contains jω as a factor. Thus if the systemdesign requires the tracking of frequency step
with zero steady-state error, the loop filter design
must contain an approximation to a perfectintegrator.
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5.2 Carrier-phase estimation with a phase-locked loop
5.2.2 Performance in Noise
Consider the input of the PLL include norrowband additive
Gaussian noise n(t), the normalize input becomes
0
0 0 0
( ) cos( ) ( )
cos( ) ( )cos ( )sinc s
r t t n t
t n t t n t t
ω θ
ω θ ω ω
= + +
= + + −
The output of the phase detector can be written as
( ) ( ) ( )
ˆ ˆ ˆsin( ) ( )cos ( )sin ( )c s
e t x t r t
n t n t θ θ θ θ
=
= − + + + 二倍频分量
As before, the loop filter eliminates the twice-carrier-
frequency terms. Denoting the second and third terms as
ˆ ˆ'( ) ( )cos ( )sinc sn t n t n t θ θ = +
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5.2 Carrier-phase estimation with a phase-locked loop
The variance of n’(t) is identical to the variance of n(t).
This variance will be denoted by 2
nσ
Consider the autocorrelation function of n’(t){ }
{ } { }
{ } { }
1 2 1 2
2 2
1 2 1 2
1 2 1 2
( , ) '( ) '( )
ˆ ˆ( ) ( ) cos ( ) ( ) sin
ˆ ˆ( ( ) ( ) ( ) ( ) )sin cos
c c s s
c s s c
R t t E n t n t
E n t n t E n t n t
E n t n t E n t n t
θ θ
θ θ
=
= +
+ +
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
2 2
2 2
0 0
0 0
ˆ ˆcos sin
ˆ ˆcos sin
c s
c s
s c n n
n n
R R R
G G G
G G G G
G G G
τ τ θ τ θ
ω ω θ ω θ
ω ω ω ω ω ω
ω ω ω ω ω
= +
= +
= = − + +
= − + +
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g g, g y
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5.2 Carrier-phase estimation with a phase-locked loop
Thus, if the noise process is white and the loop is
successfully tracking the input phase, the phase
variance is given by
2ˆ 02 L N Bθ σ =
The phase variance is a measure of the amount of jitter
of wobble in the VCO output due to noise at the input.
Here it highlight one of the many tradeoffs incommunication theory.
Clearly, one wish the phase variance to be small, which
for a given noise level implies a small loop bandwidth, B L ,which implies a narrow H( ω ). However, the narrower the
effective bandwidth of H( ω ), the poorer will be the loop’s
ability to track incoming signal phase changes.
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5.2 Carrier-phase estimation with a phase-locked loop
Squaring Loop
For DSB-SC AM signals, the received signal r(t) does not
contains dc component. So we cannot extract a carrier-
signal component directly from r(t)
If we square r(t)
2 2 2 2
2 2 2 2 21 12 2
( ) ( )cos (2 )
( ) ( )cos (4 2 )
c c c
c c c c
r t A m t f t noiseterms
A m t A m t f t noiseterm
π φ
π φ
= + += + + +
Since m 2(t)>0, there is signal power at the frequency 2f c,
which can be used to drive a phase-locked loop.
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5.2 Carrier-phase estimation with a phase-locked loop
Square-law
device
Bandpass
filter tuned
to 2f c
Loop
filter
VCO
÷2
r(t) r 2(t)
cos(4π f ct+2Φ)
e((t)
ˆsin(4 2 )c
f t π φ +
ˆsin(2 )c f t π φ +
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5.2 Carrier-phase estimation with a phase-locked loop
Costas Loop.
Lowpass
filter
VCO Loopfilter
Lowpass
filter
90o
-phaseshift
ˆsin(2 )c f t π φ +
ˆcos(2 )c f t π φ +
( )e t ( )s t
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Summarizing
Effect of Noise on Analog Communication
Systems (required)
Carrier-Phase Estimation with a Phase-
Locked Loop (PLL, general learn)
思考题:
1、等效基带系统是如何定义的。
2、
DSB,
SSB,
Conventional AM三种调制方式
的抗噪声性能如何?
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What is the next
Chapter5
Effect of noise on angle modulation (Section 5.3)
Comparison of analog-modulation systems(Section 5.4)
Effects of transmission losses and noise in analog
communication systems (Section 5.5)
Homework: 5.4,5.5, 5.8
Note: Homework is due to one week after it is
assigned.
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Chapter 5
Thank you for your attention!
Any question?
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5.3 Effect of Noise on Angle Modulation
Figure 5.3.1 Block diagram of receiver for a general angle-demodulated signal
BPF Limiter Discriminator LPF( )u t
( )r t ( ) y t
demodulator
( )n t
In the block diagram of the angle demodulation, the bandwidthof the BPF is B c= 2(
+1)W , where
is the deviation ratio and
W is the bandwidth of the message signal. So the bandwidth of
the LPF is W .( ) cos[2 ( )]
( )( )
2 ( )
c c
p n
d n
u t A f t t
K m t PM t
f m d FM
π θ φ
φ π α α
= + +
⎧⎪= ⎨
⎪⎩ ∫
2
0
2
2
f d
cb
K f
SNR
N B
π =
=
where
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5.3 Effect of Noise on Angle Modulation
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( )
( )
2 2
cos 2 sin 2
cos(2 ( )) cos(2 arctan )
cos(2 ( )) cos(2 ( ))
( ) cos(2 ( ))
c c s c
s
c c c s c
c
c c n c n
c
r t u t n t
u t n t f t n t f t
n t A f t t n t n t f t
n t
A f t t V t f t t
R t f t t
π π
π φ π
π φ π φ
π ϕ
= +
= + −
= + + + +
= + + +
= +
The output of the demodulator would beψ
(t). For larger SNR,
we can see ( ) ( )
( )
( )
( )
( ) sin ( ) ( )( ) arctan
( ) cos ( ) ( )
e
n n
e
c n n
t t t
V t t t t
A V t t t
ϕ φ φ
φ φ φ
φ φ
= +
−=
+ −
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5.3 Effect of Noise on Angle Modulation
r(t)
Ac
Φ(t) Φn(t)
V n(t)
Re
Im
Φe(t)
Θ=0
Figure 5.3.2 Phasor diagram for angle demodulation, assuming SNRT>>1
Ψ (t)
r(t)
Ac
Φ(t)Φn(t)
V n(t)
Re
Im
Φe(t)
Θ=0
Figure 5.3.3 Phasor diagram for angle demodulation, assuming SNRT<1
Ψ (t)
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5.3 Effect of Noise on Angle Modulation
For small SNR, we have
The output signal of the demodulator is
[ ][ ]
[ ]
1
( ) ( ) ( )
sin ( ) ( )( ) tan( ) cos ( ) ( )
sin ( ) ( )( )
n e
c ne
n c n
c
nn
t t t
A t t t V t A t t
A
t t V t
ϕ φ φ
φ φ φ φ φ
φ φ
−
= +
−=+ −
≈ −
( )
( ) 1 ( )
2
D
D
PM K t
y t d t K
FM dt
ϕ
ϕ
π
⎧⎪
= ⎨⎪⎩
Where K D is the discriminator constant, where we suppose K D=1.
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5.3 Effect of Noise on Angle Modulation
( ) ( ) ( ) ( )
( )( )
( )
( )
( )sin ( ) ( ) ( )
( ),
2 ,
( ) ( ),( ) 1
( ) ( ),2
( )( ) sin ( ) ( ) ,
( )1
( ) sin ( ) ( ) ,2
nn n
c
p
t
f
p n
f n
n p n
c
n
f nc
V t t t t t t Y t
A
K m t PM t
k m d FM
K m t Y t PM y t d
K m t Y t FM dt
V t K m t t t PM A
V t d
K m t t t FM dt A
ϕ φ φ φ φ
φ π τ τ
π
φ φ
φ φ π
−∞
≈ + − = +
⎧⎪= ⎨
⎪⎩
+⎧⎪= ⎨
+⎪⎩
⎧ + −⎪⎪
= ⎨⎡ ⎤⎪ + −⎢ ⎥⎪ ⎣ ⎦⎩
∫
1. For a large SNR
Noise at the output
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5.3 Effect of Noise on Angle Modulation
Let us study the properties of the noise component
given by
Consider the noise component at the output of the
demodulator, for simplicity, supposeφ (t)=0, then
The output noise power spectral density is
( ) ( )
( )sin ( ) ( )n
n n
c
V t Y t t t
A
φ φ = −
( ) ( ) ( )( ) sin[ ( ) ( )] sin ( )n n sn n n
c c c
r t r t n t Y t t t t A A A
φ φ φ = − = =
02
2 2
0 02 2 2
1
( )1 1
(2 )(2 )
c
n
c c
N PM A
S f
f N f N FM A Aπ π
⎧⎪⎪
= ⎨⎪ =⎪⎩
Here, we use
2
( ) /
( ) (2 ) ( ) y x
y t dx dt
S f f S f π
=
⇔ =
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5.3 Effect of Noise on Angle Modulation
-W W
02
1
c
N A
f
( )nPS f
Bc-Bc
Figure 5.3.4 The Noise Spectrum of PM
Figure 5.3.5 The Noise Spectrum of FM
-W W
2
02
1
c
f N A
f
( )nF S f
Bc-Bc
Noise affect the
signal
Noise affect the
signal
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5.3 Effect of Noise on Angle Modulation
For│
f │
<W, the spectrum of the noise components in
the PM and FM are given by
0
2
0 0
2
2( ) c
c
N
An N
A
PM S f FM f
⎧⎪= ⎨⎪⎩
For FM system, the effect of noise for higher-frequencycomponents is much higher than the effect of noise on
lower-frequency components.
The noise power at the output is:0
2
3
0 2
2
2
3
( ) c
o o
c
WN
W A
n n N W W
A
PM P S f df
FM −
⎧⎪
= = ⎨
⎪⎩
∫
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5.3 Effect of Noise on Angle Modulation
As we know the output of the demodulator is given by
( )
( )12
( ) sin( ( ) ( )),
( ) ( ) sin( ( ) ( )),
n
c
n
c
V t
p n A
V t d f ndt A
k m t t t PM
y t FM k m t t t π
φ
φ
⎧ + Φ −⎪=
⎨ + Φ −⎪⎩So the output signal power is
2
2o
p M
s
f M
k P PM P
k P FM ⎧⎪= ⎨⎪⎩
The SNR becomes
( ) ( )
( ) ( )
2 2 2
20
2 2 2
2 20
2 max ( )
3
2 max ( )3
p c M p M
f c M f M
k A PP S N W N bm t
k A PP S
N W N bW m t
PM S
FM N
β
β
⎧ =⎪⎛ ⎞
= ⎨⎜ ⎟⎝ ⎠
⎪ =⎩
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5.3 Effect of Noise on Angle Modulation
where
max ( )
max ( )
f
p p
k m t
f W
k m t PM
FM
β
β
⎧ =⎪⎨
=⎪⎩
Note that ( )2
max ( )
M
n
P
M m t
P= is the average-to peak-power ration of
the message signal (or, equivalently the power content of the
normalized message). Therefore
( ) ( ) ( )
( ) ( ) ( )
2
2
212
max ( )
212
max ( )3 3
n
n
S S p M M N N m t b b
o S S f M M N N m t b b
P PPM S
FM N P P
β
β
Ω
Ω
−
−
⎧=⎪⎛ ⎞
= ⎨⎜ ⎟⎝ ⎠ ⎪ =⎩
where 2( 1)c B
W
β Ω = = + is defined as the bandwidth expansion
factor
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5.3 Effect of Noise on Angle Modulation
1. In both PM and FM, the output SNR is Proportional to
the square of the modulation index . Therefore,
increasing increases the output SNR.
2. The increase in the received SNR is obtained by
increasing the bandwidth. Therefore, angle modulation
provides a way to trade-off bandwidth for transmittedpower.
3. Although we can increase the output SNR by increasing
, but increasing will cause threshold effect, thesignal will be lost in the noise.
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5.3 Effect of Noise on Angle Modulation
4. Increasing the transmitter power will increase the
output SNR. In AM, any increase in the received
power directly increases the signal power at the output
of the receiver. In angle modulation what increases the
output SNR is a decrease in the received noise power.
5.
In FM, the effect of noise is higher at higherfrequencies. This means that signal components at
higher frequency will suffer more from noise than the
lower frequency components.
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5.3 Effect of Noise on Angle Modulation
5.3.1 Threshold Effect in Angle Modulation
Threshold effect: at low SNRs, signal and noise
components are so intermingled that one can not
recognize the signal from the noise, a mutilation or
threshold effect is present.
The existence of the threshold effect places an upper-limit
on the trade-off between bandwidth and power in an FMsystem.
It can be shown that at threshold the following approximate
relation between baseband SNR and f holds in an FMsystem:
, 0
20( 1) R
b th
S P
N N W β
⎛ ⎞= = +⎜ ⎟
⎝ ⎠
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5.3 Effect of Noise on Angle Modulation
In general, there are two factors that limit the valueof the modulation index .
1. One is the limitation on the channel bandwidth which
affects through Carson’s rule B c=2(
+1).2. The other is the limitation on the received power that
limits the value of to less than what is derivedfrom the upper equation.(seeing pp245, Figure 5.16)
3. If we want to employ the maximum availablebandwidth, we using equation
( )260 1n M
o
S P
N
β β ⎛ ⎞
= +⎜ ⎟
⎝ ⎠
Using the threshold relation, we determine the requiredminimum received power to make the whole allocated
bandwidth usable.
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5.3 Effect of Noise on Angle Modulation
Example
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5.3 Effect of Noise on Angle Modulation
5.3.2. Pre-emphasis and De-emphasis Filtering
Basic ideas:
1. Pre-emphasis filter: is a filter that at low frequencies
does not affect the signal and at high frequencies acts asa differentiator. A highpass filter is a goodapproximation to such a system.
2. De-emphasis filter: is a filter that at low frequencies has
a constant gain and at high frequencies behaves as anintegrator. A lowpass filter is a good approximation tosuch a system.
3. Another way to understand emphasis is from the part
of noise.Example of Pre-emphasis and De-emphasis Filtering:
0
1( )
1
d f
f
H f
j
=
+where 6
10 2 75 10
2100 f Hzπ × ×
= ≈
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5.3 Effect of Noise on Angle Modulation
Analyzing the effect of pre-emphasis and de-emphasisfiltering on the overall SNR in FM broadcasting.
The only filter that has an effect on the received noise is the
receiver noise is the receiver filter that shapes the power-spectral density of the noise within the messagebandwidth.
The noise power-spectral density after de-emphasis filter is
2
20
2 20
2
1( ) ( ) ( )
1onPD n d f
c f
N S f S f H f f
A= =
+The noise power is
30 0
2
0 0
2( ) arctan
W
nPD nPD
cW
N f W W P S f df
A f f −
⎡ ⎤= = −⎢ ⎥⎣ ⎦
∫
( )( ) ( )0
0 0
3
13 arctan
W S f
N oPDS W W
N f f o
= −
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5.4 Comparison of Analog-modulation systems
Bandwidth Efficiency
SSB-SC→
VSB→
FM
Power Efficiency.
FM→conventional AM→VSB+Carrier
Ease of Implementation
conventional AM→
VSB+C→
FM Noting:
1. SSB-SC and DSB-SC never used for broadcasting
purposes.2. DSB-SC is hardly used in practice.
School of Information Science and Engineering, Shandong University5.5 Effects of Transmission Losses and Noise in Analog
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Communication System
Transmitted
Signal
s(t)
Attenuation
α
Noise
n(t)
Received signal
r(t)=αs(t)+n(t)
channel
Mathematical model of channel with attenuation and additive noise
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Communication System
5.5.1 Characterization of Thermal Noise Sources
5.5.2 Effective Noise Temperature and Noise Figure
5.5.2 Transmission Losses
The amount of signal attenuation generally depends on the
physical medium, the frequency of operation, and the
distance between the transmittrer and the receiver.
Defined the loss L =PT /P R
1. In wireline channels, the transmission loss is usually
given in terms of dB per unit length
2. In LOS radio systems the transmission loss is given as
24 d π
λ
⎛ ⎞=
⎜ ⎟⎝ ⎠L
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Communication System
5.5.4 Repeaters for Signal Transmission
Analog repeaters are basically amplifiers that generally
used in telephone wireline channels and microwave LOS
radio channels to boost the signal level and , thus , to
offset the effect of signal attenuated by the lossy
transmission medium.
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S i i
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Summarizing
Effect of noise on angle modulation
Comparison of analog-modulation systems
Effects of transmission losses and noise in
analog communication systems
思考题
:
1、
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Wh t i th t
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What is the next
Chapter 6
Modeling of Information Source
(Section 6.1)
Source-Coding Theorem (Section 6.2)
Homework: 5.7 5.9 5.13
Note: Homework is due to one week after it
is assigned.
School of Information Science and Engineering, Shandong University
Ch t 5