Chapter 5course.sdu.edu.cn/Download2/20150625110055202.pdf · 2015. 6. 25. · cn c s cn c s cc n cc cs n cs. Vt A am t n t n t A am t n t n t A n t am t An t nt nt amt nt nt =+ +

School of Information Science and Engineering, Shandong University

Principles of the Communications

Chapter 5Chapter 5

Effect of Noise on Analog Effect of Noise on Analog Communication SystemsCommunication Systems

Copyrights Zhu, Weihong



Chapter 5 Contents

5.1 Effect of noise on linear-modulation systems5.2 Carrier-phase estimation with a phase-locked loop(PLL).5.3 Effect of noise on angle modulation5.4 Comparison of analog-modulation systems5.5 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 )cf tπ φ+

( )y t ( )oy t

( ) ( ) ( )m ir t u t n t= +

( ) ( )cos2 ( )sin 2i c c s cn t n t f t n t f tπ π= −





cf W+cfcf−cf W− −

BPF-USSB

f

cfcf− cf W+cf W−cf W− +cf 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

N0 /2

W/2-W/2

Figure 5.1.3 Baseband system





002o

W

n W

NP df N W−

= =∫

The power of the noise is

If we denote the received power by PR , the baseband SNR is given by

0

R

b

S PN 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 tπ φ= +

[ ]

[ ]

( ) ( ) cos(2 ) ( ) cos 2 ( )sin 2( ) ( ) cos(2 )

( )cos(2 ) ( ) cos 2 ( )sin 2 cos(2 )1 1( )cos( ) ( ) cos(4 )2 21 ( )cos ( )sin212

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 ty 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 ππ φπ 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.

214

104

2 2

0 0

0

2

12 2

o

c mo

oDSB n

c m c m

R

b

A PPSN P WN

A P A PWN N WP S

N W N

⎛ ⎞ = =⎜ ⎟⎝ ⎠

= = ×

⎛ ⎞= = ⎜ ⎟⎝ ⎠

2

0 0

1 1 14 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 tπ 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 tr t A m t f t n t f t n t f ty t r t f t

A m t f t A m t f tn t f t n t f t

π ππ φ π π

π φπ φ π φπ π

= + + ± −

= + + −

= +

+ ± +⎡ ⎤= ⎢ ⎥+ −⎣ ⎦

[ ][ ]

[ ]

1 1 12 2 2

1 12 2

12

12

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 tA 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

Actrually 2

02c m

oDSB

S A PN WN

⎛ ⎞ =⎜ ⎟⎝ ⎠

2

0

c m

oSSB

S A PN 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 SN P N W N W N

= × =

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

21 1 14 4 4o co c m n n n

P A P P P P= = =





5.1.4 Effect of Noise on Conventional AM

( ) [1 ( )]cos2m c n cu t A am t f tπ= +

( ) [ [1 ( )] ( )]cos 2 ( )sin 2c n c c s cr t A am t n t f t n t f tπ π= + + −

If the demodulator is a coherent demodulator1( ) { [1 ( )] ( )}2o c n c

y t A am t n t= + +

2 214 no c m

P A a P= 02nP WN=

1 14 4o cn n n

P P P= =2

2[1 ]2 n

cR m

AP a P= +





2 2 2 214

104 2

n nc m c m

oAM n

A a P A a PSN P N W

⎛ ⎞ = =⎜ ⎟⎝ ⎠

2 222

20

1

1

c

nn

n

Amm

m

a Pa Pa P N W

⎡ ⎤+⎣ ⎦=+

2

201

n

n

m R

m

a P Pa P N W

=+

b

SN

η ⎛ ⎞= ⎜ ⎟⎝ ⎠

2

21n

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 cr t A am t n t f t n t f tπ π= + + −

[ ]{ }2 2( ) 1 ( ) ( ) ( )c n c sV t A am t n t n t= + + +





1. High input SNR or high PR, , which means

[ ]{ }( ( )) 1 ( )s c nP n t A am t+So [ ]{ }( ) 1 ( ) ( )c n cV t A am t n t≈ + +Here SNR is equal to the coherent demodulator

2 2

02nc m

oAM

A a PSN N W

⎛ ⎞ =⎜ ⎟⎝ ⎠ b

SN

η⎛ ⎞= ⎜ ⎟⎝ ⎠

2. Low input SNR or low PR

[ ]{ }( ( )) 1 ( )c nP n t A am t+





[ ]{ }( ) ( )

( ) ( )

2 2

22 2 2

2 22 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 tn t n t am tn t n t

= + + +

= + + + + +

⎡ ⎤≈ + + +⎢ ⎥+⎣ ⎦

[ ]( )( ) 1 ( )( )

c cn n

n

A n tV t am tV t

≈ + +

Here the signal component is multiplied by noise and is no longer distinguishable. In this case, no meaningful SNR can be defined. It is said that this system is operating below the threshold (门限).





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 the required transmitter power and channel bandwidth if the following modulation schemes are employed?

1. DSB AM2. SSB AM3. 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

( )812 40

0.5 102 10 10

R RR

b

P PS PN N W −

⎛ ⎞ = = = ×⎜ ⎟ × ×⎝ ⎠

810 lg 80 102

T TR T

bR

P PSP PP N

− ⎛ ⎞= ⇒ = ⇒ =⎜ ⎟⎝ ⎠

550 10 2002

2 20

TT

O R

PS S dB P KWN N

BW W KHz

⎛ ⎞ ⎛ ⎞= = = ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

∼





2. For SSB AM,

3. For conventional AM, with a=0.8,

550 10 2002

10

TT

O R

PS S dB P KWN N

BW W KHz

⎛ ⎞ ⎛ ⎞= = = ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

∼

( )( ) ( )

2 2

22 2

5

2

0.8 16 / 360.22 where

1 1 0.8 16 / 36 max ( )

0.22 10 909 2 202

n

n

n

T

O b

m mm

m

TT

O

PS SN N

a P PPa P m t

PS P KW BW W KHzN

η η

η

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= = ≈ =+ +

⎛ ⎞ ≈ = ⇒ ≈ = =⎜ ⎟⎝ ⎠




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 detectorLoop filter

Voltage-controlled oscillator

Figure 5.2.1 Schematic of the basic 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.





A VCO is an oscillator whose output frequency is a linear function of its input voltage over some range of input and output. A positive input voltage will cause the VCO output frequency to be greater than its uncontrolled value, f0, while a negative voltage will cause it to be less.Phase lock is achieved by feeding a filtered version of the phase difference between the incoming signal r(t) and the output 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 tπ θ= +

Where f0 is the nominal carrier frequency and θ(t) is a slowly varying phase.





Consider a normalized VCO output of the form

0ˆ( ) 2sin 2 ( )x t f t tπ 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 tθ θ θ θ= − ≈ −





If we make the assumption that fo is the uncontrolled frequency of the VCO, we can express the difference in the VCO output frequency from f0 as the time differential of the phase term . The output frequency of the VCO is a linear function of the input voltage. Therefore, since an input voltage of zero produces an output frequency of f0, the difference in the output frequency from f0 will be proportional to the value of the input voltage y(t), or

ˆ( )tθ

0 0

1 ˆ( ) ( ) ( )2

ˆ( ) ( ) ( ) ( ) ( )

df t t Ky tdt

K e t f t K t t f t

θπ

θ θ

⎡ ⎤Δ = =⎢ ⎥⎣ ⎦⎡ ⎤= ∗ ≈ − ∗⎣ ⎦

Where K0 /2п is the gain of the VCO.





Consider the Fourier transform of the upper equation

( ) ( ) ( )0ˆ ˆ( )j K Fωθ ω θ ω θ ω ω⎡ ⎤= −⎣ ⎦( )( )

( )( )

0

0

ˆ( )

K FH

j K Fθ ω ω

ωθ ω ω ω

= =+

Steady-state tracking characteristics (稳态跟踪特征)

( )( )E e tω = ⎡ ⎤⎣ ⎦F( ) ( )

[ ] ( )( )( )0

ˆ

1 ( )H

jj K F

θ ω θ ω

ω θ ω

ωθ ωω 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 Eω

ω ω→∞ →

=

( ) ( )( )

2

00

lim ( ) limt j

je t

j K Fωω 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 StepConsider a loop’s steady-state response to a phase

step at the loop input.





Solution Assuming that the PLL was originally in phase lock, a phase

step will throw the loop out of lock. Having abruptly changed, however, the input phase again becomes stable. This should be the easiest type of phase disturbance for a PLL to deal with. The Fourier transform of a phase step will be taken to be

( ) ( ( ))u tjφθ ω φωΔ

= Δ =F

Where Δ Φ is the magnitude of the step. So( ) ( )

( ) ( )

2

0 00 0

lim ( ) lim lim 0t j j

j je tj K F j K Fω ωω 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 has a nonzero dc response.





☺Example 2. Response to a Frequency StepNext, consider a loop’s steady-state response to a frequency

step at the input.Solution

Since phase is the integral of frequency, the input phase will change linearly as a function of time for a constant input- frequency offset. The Fourier transform of the phase characteristic will be the transform of the integral of the frequency characteristic. Since the frequency characteristic is a step, and the transform of an integral is the transform of the integrand divided by the parameter jω, it follows that

( ) 2( )jωθ ωωΔ

=

Where Δω 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 Fω ωω θ ω ω ω

ω ω ω ω→∞ → →Δ Δ

= = =+ +

( ) 11

Fjωωω ω

=+

If it is a lead –lag(超前滞后) filter, then ( ) 1 22 1

jFj

ω ω ωωω ω ω⎛ ⎞ +

= ⎜ ⎟ +⎝ ⎠so

0

lim ( )t

e tKω

→∞

Δ=





This steady-state error will exist regardless of the order of the filter, unless the denominator of F(ω), contains j ω as a factor. Thus if the system design requires the tracking of frequency step with zero steady-state error, the loop filter design must contain an approximation to a perfect integrator.





5.2.2 Performance in NoiseConsider 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 tt n t t n t tω 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 tθ 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 tθ θ= +





The variance of n’(t) is identical to the variance of n(t). This variance will be denoted by 2nσ

Consider the autocorrelation function of n’(t)

{ }{ } { }{ } { }

1 2 1 2

2 21 2 1 2

1 2 1 2

( , ) '( ) '( )ˆ ˆ( ) ( ) cos ( ) ( ) sin

ˆ ˆ( ( ) ( ) ( ) ( ) )sin cosc 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 sinˆ 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

= +

= +

= = − + +

= − + +





For the special case of white noise, we have Gn(ω)=N0/2, where N0 is the single-sided spectral density of the white noise. So

G(ω)=N0( ) ( ) ( ) 2ˆG G Hθ ω ω ω=so

The variance of the output phase is then

( ) ( ) ( )2 22 0ˆ1

2 2NG H d H d

θσ ω ω ω ω ω

π π∞ ∞

−∞ −∞= =∫ ∫

Let ( ) 212

2L LW B H dω ω

π∞

−∞= = ∫

WL is called the two-sided loop bandwidth, BL is the single-sided loop bandwidth.





Thus, if the noise process is white and the loop is successfully tracking the input phase, the phase variance is given by

2ˆ 02 LN Bθσ =

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 in communication theory.Clearly, one wish the phase variance to be small, which for a given noise level implies a small loop bandwidth, BL , 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.





Squaring LoopFor 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 m2(t)>0, there is signal power at the frequency 2fc , which can be used to drive a phase-locked loop.





Square-law device

Bandpass filter tuned

to 2fc

Loop filter

VCO

÷2

r(t) r2(t)

cos(4πfc t+2Φ)

e((t)

ˆsin(4 2 )cf tπ φ+

ˆsin(2 )cf tπ φ+





Costas Loop.

Lowpass

filter

VCOLoop filter

Lowpass

filter

90o-phase shift

ˆsin(2 )cf tπ φ+

ˆcos(2 )cf tπ 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

Chapter5Effect 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 bandwidth of the BPF is Bc =2(β+1)W, whereβ 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 tK m t PM

tf m d FM

π θ φ

φπ α α

= + +

⎧⎪= ⎨⎪⎩ ∫

2

0

2

2

f d

cb

K f

ASNRN B

π=

=

where





( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( )( )

2 2

cos 2 sin 2

cos(2 ( )) cos(2 arctan )

cos(2 ( )) cos(2 ( ))( ) cos(2 ( ))

c c s c

sc 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 tA f t t n t n t f t

n t

A f t t V t f t tR t f t t

π π

π φ π

π φ π φπ ϕ

= +

= + −

= + + + +

= + + +

= +The output of the demodulator would be ψ(t). For larger SNR,we can see

( ) ( )( )( )

( )

( )sin ( ) ( )( ) arctan

( )cos ( ) ( )

e

n ne

c n n

t t t

V t t tt

A V t t t

ϕ φ φ

φ φφ

φ φ

= +

−=

+ −





r(t)

Ac Φ(t) Φn(t)

Vn (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) Vn (t)

Re

Im

Φe (t)

Θ＝0

Figure 5.3.3 Phasor diagram for angle demodulation, assuming SNRT





For small SNR, we have

The output signal of the demodulator is

[ ][ ]

[ ]

1

( ) ( ) ( )sin ( ) ( )

( ) tan( ) cos ( ) ( )

sin ( ) ( )( )

n e

c ne

n c n

cn

n

t t tA t t

tV t A t t

A t tV t

ϕ φ φφ φ

φφ φ

φ φ

−

= +

−=

+ −

≈ −

( )( ) 1 ( )

2

D

D

PMK ty t d tK

FMdt

ϕϕ

π

⎧⎪= ⎨⎪⎩

Where KD is the discriminator constant, where we suppose KD =1.





( ) ( ) ( ) ( )

( )( )

( )

( )

( ) sin ( ) ( ) ( )

( ),

2 ,

( ) ( ),( ) 1( ) ( ),

2( )( ) sin ( ) ( ) ,

( )1( ) sin ( ) ( ) ,2

nn n

c

pt

f

p n

f n

np n

c

nf n

c

V tt t t t t Y tA

K m t PMt

k m d FM

K m t Y t PMy t dK m t Y t FM

dtV tK m t t t PM

A

V tdK m t t t FMdt A

ϕ φ φ φ φ

φπ τ τ

π

φ φ

φ φπ

−∞

≈ + − = +

⎧⎪= ⎨⎪⎩

+⎧⎪= ⎨

+⎪⎩⎧ + −⎪⎪= ⎨ ⎡ ⎤⎪ + −⎢ ⎥⎪ ⎣ ⎦⎩

∫

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, suppose φ (t)=0, then

The output noise power spectral density is

( ) ( ) ( )sin ( ) ( )nn nc

V tY t t t

Aφ φ= −

( ) ( ) ( )( ) sin[ ( ) ( )] sin ( )n n sn n nc c c

r t r t n tY t t t tA A A

φ φ φ= − = =

02

2 20 02 2 2

1

( )1 1(2 )

(2 )

cn

c c

N PMA

S ff N f N FM

A Aπ

π

⎧⎪⎪= ⎨⎪ =⎪⎩

Here, we use

2

( ) /( ) (2 ) ( )y x

y t dx dtS f f S fπ=

⇔ =





-W W

02

1

c

NA

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

202

1

c

f NA

f

( )nFS f

Bc-Bc

Noise affect the signal

Noise affect the signal





For │f│





As we know the output of the demodulator is given by ( )

( )12

( ) sin( ( ) ( )),( )

( ) sin( ( ) ( )),

n

c

n

c

V tp nA

V tdf ndt A

k m t t t PMy t

FMk m t t tπ

φ

φ

⎧ + Φ −⎪= ⎨+ Φ −⎪⎩

So the output signal power is2

2op M

sf M

k P PMP

k P FM⎧⎪= ⎨⎪⎩

The SNR becomes

( ) ( )

( ) ( )

2 2 2

20

2 2 2

2 20

2 max ( )

3

2 max ( )3

p c M pM

f c M fM

k A PP SN W N bm t

k A PP SN W N bW m t

PMSFMN

β

β

⎧ =⎪⎛ ⎞ = ⎨⎜ ⎟

⎝ ⎠ ⎪ =⎩





where

max ( )

max ( )f

p p

k m tf W

k m t PMFM

β

β

⎧ =⎪⎨

=⎪⎩

Note that ( )2max ( )M

n

PMm 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

212max ( )

212max ( )3 3

n

n

S Sp M MN Nm tb b

o S Sf M MN Nm tb b

P P PMSFMN P P

β

β

Ω

Ω

−

−

⎧ =⎪⎛ ⎞ = ⎨⎜ ⎟⎝ ⎠ ⎪ =

⎩where 2( 1)cB

WβΩ = = + 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 β. Therefore, increasing β 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 transmitted power.

3. Although we can increase the output SNR by increasing β, but increasing β will cause threshold effect, the signal 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 higher frequencies. 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 ModulationThreshold 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 FM system.

It can be shown that at threshold the following approximate relation between baseband SNR and βf holds in an FM system:

, 0

20( 1)Rb th

S PN N W

β⎛ ⎞ = = +⎜ ⎟⎝ ⎠





In general, there are two factors that limit the value of the modulation index β.

1. One is the limitation on the channel bandwidth which affects β through Carson’s rule Bc =2(β+1).

2. The other is the limitation on the received power that limits the value of β to less than what is derived from the upper equation.(seeing pp245, Figure 5.16)

3. If we want to employ the maximum available bandwidth, we using equation

( )260 1nM

o

S PN

β β⎛ ⎞ = +⎜ ⎟⎝ ⎠

Using the threshold relation, we determine the required minimum received power to make the whole allocated bandwidth usable.





Example





5.3.2. Pre-emphasis and De-emphasis FilteringBasic ideas: 1. Pre-emphasis filter: is a filter that at low frequencies

does not affect the signal and at high frequencies acts as a differentiator. A highpass filter is a good approximation 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 an integrator. A lowpass filter is a good approximation to such a system.

3. Another way to understand emphasis is from the part of noise.

Example of Pre-emphasis and De-emphasis Filtering:

0

1( )1d ff

H fj

=+

where 610 2 75 10 2100f Hzπ× ×= ≈





Analyzing the effect of pre-emphasis and de-emphasis filtering 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 message bandwidth.

The noise power-spectral density after de-emphasis filter is

2

20

2 202

1( ) ( ) ( )1onPD n d fc f

NS f S f H f fA

= =+

The noise power is 3

0 02

0 0

2( ) arctanW

nPD nPDcW

N f W WP S f dfA f f−

⎡ ⎤= = −⎢ ⎥

⎣ ⎦∫

( )( )

( )00 0

3

13 arctan

WSfN oPD

S W WN f fo

=−




5.4 Comparison of Analog-modulation systems

Bandwidth EfficiencySSB-SC→VSB→FMPower Efficiency.FM→conventional AM→VSB+CarrierEase of Implementationconventional AM→VSB+C→FMNoting:

1. SSB-SC and DSB-SC never used for broadcasting purposes.

2. DSB-SC is hardly used in practice.




5.5 Effects of Transmission Losses and Noise in Analog Communication System

Transmitted Signal

s(t)

Attenuation α

Noisen(t)

Received signal r(t)=αs(t)+n(t)

channel

Mathematical model of channel with attenuation and additive noise





5.5.1 Characterization of Thermal Noise Sources5.5.2 Effective Noise Temperature and Noise Figure5.5.2 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 /PR1. 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 dπλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

L





5.5.4 Repeaters for Signal TransmissionAnalog 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.




Summarizing

Effect of noise on angle modulationComparison of analog-modulation systemsEffects of transmission losses and noise in analog communication systems

思考题:

1、




What is the next

Chapter 6Modeling 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.




Chapter 5

Thank you for your attention!

Any question?

Chapter 5Chapter 5 Contents5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.1 Effect of Noise on Linear-modulation Systems5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loop5.2 Carrier-phase estimation with a phase-locked loopSummarizingWhat is the nextChapter 55.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.3 Effect of Noise on Angle Modulation5.4 Comparison of Analog-modulation systems5.5 Effects of Transmission Losses and Noise in Analog Communication System 5.5 Effects of Transmission Losses and Noise in Analog Communication System5.5 Effects of Transmission Losses and Noise in Analog Communication SystemSummarizingWhat is the nextChapter 5

Chapter 5course.sdu.edu.cn/Download2/20150625110055202.pdf · 2015. 6. 25. · cn c s cn c s cc n cc cs n cs. Vt A am t n t n t A am t n t n t A n t am t An t nt nt amt nt nt =+ +

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