EFFECT OF NOISE IN ANALOG COMMUNICATION SYSTEMS

7/21/2019 EFFECT OF NOISE IN ANALOG COMMUNICATION SYSTEMS

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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



Copyrights Zhu, Weihong



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






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 = −







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







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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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






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

+ + − + +

= + =






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

= = =

= × =






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?






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

+

= − + − + + +

+ + + + +

+ + + − + +

= + =






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= = =






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= +






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







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







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







[ ]{ }

( ) ( )

( ) ( )

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







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








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

⎛ ⎞ ⎛ ⎞= = = ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

= =

∼








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






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

School of Information Science and Engineering Shandong University







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.








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.








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 = − ≈ −








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.








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








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.








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.








☺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








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=








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.





Sc oo o o at o Sc e ce a d g ee g, S a do g U ve s ty



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 = +





g g, g y



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

= +

= +

= = − + +

= − + +




g g, g y



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.





g g, g y



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.




Copyrights Zhu, Weihong Principles of the Communications


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 +






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





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？





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.





Chapter 5

Thank you for your attention!

Any question?





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






( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )

( )

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

= +

−=

+ −






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)






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.






( ) ( ) ( ) ( )

( )( )

( )

( )

( )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






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

=

⇔ =






-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






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 −

⎧⎪

= = ⎨

⎪⎩

∫






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

⎧ =⎪⎛ ⎞

= ⎨⎜ ⎟⎝ ⎠

⎪ =⎩






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






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.






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.






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

⎛ ⎞= = +⎜ ⎟

⎝ ⎠






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.






Example






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 × ×

= ≈






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

= −





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




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


5.5 Effects of Transmission Losses and Noise in Analog





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


5.5 Effects of Transmission Losses and Noise in Analog





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.


S i i




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、


Wh t i th t




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.


Ch t 5



Chapter 5

Thank you for your attention!

Any question?

EFFECT OF NOISE IN ANALOG COMMUNICATION SYSTEMS

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