Chapter 10: Noise in Analog Modulation A. Bruce …homepages.wmich.edu/~bazuinb/ECE4600/Ch10_1.pdf6...

Communication Systems, 5e

Chapter 10: Noise in Analog Modulation

A. Bruce CarlsonPaul B. Crilly

© 2010 The McGraw-Hill Companies

Chapter 10: Noise in Analog Modulation

• Bandpass Noise• Linear CW Modulation With Noise• Exponential CW Modulation With Noise• Comparison Of CW Modulation Systems• Phase-locked Loop Noise Performance• Analog Pulse Modulation With Noise

© 2010 The McGraw-Hill Companies

3

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Model of a CW communication system with noise: Figure 10.1-1

CW Communication with Noise

ttf2cosLtAtx cc

tnttf2cosLtAtv c

thtnttfLtAtPreD Rc

2cos

ttf2costAtx c

4

Signal and Noise Power

• What are the signal and noise powers at the receiver?

• What is the receiver input power

2T

2T

Ts dttxtxT1limP

2n tnEP

2T

2T

ccT

2T

2T

Tv dttntxtntxET1limdttvtvE

T1limP

Noise: a zero mean random variable

Power vs EnergyAs we are eventually performing a ratio to define

SNR, energy is often used instead of power.

5

Receiver Signal plus Noise Power

• What is the receiver input power

2T

2T

ccT

2T

2T

Tv dttntxtntxET1limdttvtvE

T1limP

2T

2T

2c

2cTv dttntntx2txE

T1limP

2T

2T

2c

2cTv dttnEtnEtx2tx

T1limP

2

2

22

2

2

2

2 021limT

T

T

Tc

T

TcTv dttnEdttxdttx

TP

ns2

2T

2T

2cTv PPtnEdttx

T1limP

6

Signal-to-Noise Ratio (SNR)

• The SNR is a measure of the signal power to the noise power at a point in the receiver.– Typically described in dB– The above computation was performed at the input

• Matlab SNR example: SNR_AM_Example.m– Pre-D “AM” SNR based in filter Beqn

• Effective BEQN RF due to sampling spectrum (B=Fis/2)• AM signal power based on carrier plus signal

– Also run SNR_AM_Perf.m

n

s

PPSNR

7

Noise Equivalent Bandwidth

• Since the noise power spectrum is uniform, a systems average noise power is the product of the noise power and the integral of the filter power.

20NN

2NN fH

2NfSfHfS

00

0

20

20NN dffHNdffH

2N0R

8


• When filtering, it is convenient to think of band-limited noise, where the filter is a rect function with bandwidth BEQN

0

2020NN dffH2

2NdffH

2N0R

EQN2

EQNPower_DC0

2

elmod_rect0

2 B0HBGaindffHdffH

EQNPowerDCelrect B

frectGainfH2_mod_

20

2

0H

dffHBEQN

2Power_DC 0HGain

9


• Low pass filter 0Hgain_coherent

• For a unity gain filter – assumed when computing receiver input noise power

EQN0EQN0

NNN BNB22

N0RP

2Power_DC 0HGain

20

2

EQN0H

dffHB

0

2EQN dffHB

EQNEQNNNN BNHBHNRP 0220 002

20

10

Filtering

• What happens if the receiver input is filtered?

• What effect does the filter have on the signal?– None or slight band edge de-emphasis, if and only if the

filter is “wider” than the signal bandwidth– Now you know why a 3dB bandwidth isn’t that useful,

(3dB1/2 power point) in audio/RF applications!

ththtntxtv 21cf

ththtnththtxtv 2121cf

n

s

PPSNR

11

Filtering

• What effect does the filter have on the noise?– Normally you would expect for two filters

– Assume that the filters follow each other and that the first filter is narrower than the second filter

1_01_

FilterEQN

sFilterPost BN

PSNR

1_01__ FilterEQNFilterPostN BNP

1_0

2_1_02__ ,min

FilterEQN

FilterEQNFilterEQNFilterPostN

BNBBNP

2_02__ FilterEQNfilterpostN BNP 2_0

2_FilterEQN

sFilterPost BN

PSNR

1_02_

FilterEQN

sFilterPost BN

PSNR

12

Filters Provide SNR “Gain”

• If filter 2 Beq < filter 1 Beq:

• You expect the IF filter to be smaller than the front-end RF or “pre-filtering” performed

– Think about kTB at different bandwidths and you will derive the same “gain”

– In typical receivers, the IF filter sets the Pre-Demodulation Bandwidth

2Filter_EQN

1Filter_EQN1Filter_Post

2Filter_EQN0

s2Filter_Post B

BSNR

BNPSNR

2Filter_EQN

1Filter_EQNFilter B

BGain

13

Computed SNRs

• Pre-detection Signal-to-Noise Ratio– Since filtering is a normal process in a receiver, we

typically consider the SNR prior to demodulation– The effect of filtering (BPF and/or LPF) and mixing are

taken into account

• The Pre-D SNR may be significantly different than either:1) An SNR based on the signal bandwidth (textbook γ)

2) The post-demodulation SNR

WNP

0

s

FilterPostFilter B

WSNR

14

Bandpass Noise Processing

• What happens after real mixing and lowpass filtering?– assume LPF passes the entire baseband.

cfTc Bf

cfTc Bf

TBTB

20N

tf2cos c Band Pass

FilterLowPass

Filter

Bandpass filter bandwidth may not be centered on fc• an alpha offset

15


(a) General case; (b) symmetric-sideband case; (c) suppressed-sideband case: Figure 10.1-3

Lowpass PSD of the quadrature components of bandpass noise

10 or

5.0

5.00

16

Quadrature Noise (1)

• Noise in a quadrature process

• Noise power is related as

• What about ?

tf2sintntf2costntn cqci

22 2sin2cos tftntftnEtnE cqci

2

0222 NtnEtnEtnE qi

17

Quadrature Noise (2)

• Noise in a quadrature process 2

cqci2 tf2sintntf2costnEtnE

tf2sintn

tf2sintf2costntn2tf2costn

EtnE

c22

q

ccqi

c22

i2

222cos

21222cos

21 222 tftntftnEtnE cqci

22

121 0222 NtnEtnEtnE qi

Mixing Noise (1)

• Think of the two noise bands as1. The band of interest2. The image band

18

thtftfftntfftn

thtftfftntfftnthtftn

IFLOIFLOqIFLOi

IFLOIFLOqIFLOiIFLO

2cos2sin2cos

2cos2sin2cos2cos

22

11

th

tfftntfftn

tftntftn

thtfftntfftn

tftntftnthtftn

IFIFLOqIFLOi

IFqIFi

IFIFLOqIFLOi

IFqIFiIFLO

22sin22cos

2sin2cos

21

22sin22cos

2sin2cos

212cos

22

22

11

11

thtftntftn

thtftntftnthtftn

IFIFqIFi

IFIFqIFiIFLO

2sin2cos21

2sin2cos212cos

22

11

Mixing Noise (2)

• Defining the equivalent IF noise

• But this is the same as quadrature noise

• Mixing doesn’t change the noise power 19

thtftntftn

thtftntftnthtftn

IFIFqIFi

IFIFqIFiIFLO

2sin2cos21

2sin2cos212cos

22

11

thtftntftnthtn IFIFqIFiIF 2sin2cos

th

tftntn

tftntnthtftn IF

IFii

IFii

IFLO

2sin21

21

2cos21

21

2cos

21

21

22

121 02

22

12 NtnEtnEtnE iii

221

21 02

22

12 NtnEtnEtnE qqq

20

Mixing Noise to Baseband

• What if we split bandpass noise into two distinct noise bands, BT/2 above and below the carrier/IF?

• Noise power is related as

• Noise bands get added …

WNBNBNtnE TTcarrier 0002 22

2

thtfftntfftn

thtfftntfftntn

BcqBci

BcqBci

222

111

2sin2cos2sin2cos

WNBNBNtnEtnE TTBelowCAboveC 00

022

21 22

22

WNBNtnEtnEtnE T 002

22

12 2

2TBWfor

21


(a) General case; (b) symmetric-sideband case;

(c) suppressed-sideband case: Figure 10.1-3

Mixing Noise to Baseband

10 or

5.0

5.00

cfTc Bf

cfTc Bf

TBTB

20N

Mix to BasebandW=½ BT

N0BT=2N0W

Mix to IFBT

N0BT

Not Desired

Why did we do these derivations?

• The past derivations were all about mixing and filtering.– Quadrature noise is the noise that gets mixed to the

intermediate. The bandwidth and noise power do not change

– Quadrature noise is the noise that gets mixed to baseband. The bandwidth is halved and the noise power is doubled the LPF bandwidth standard noise power.

22

WNBNBNtnE TT 0002 22

2

WNBNtnE T 002 2WBT

2

23

Complex Noise

• Noise in a complex process

• Noise power is related as tnjtntn qi

222 tntntnjtntnjtnEtnE qiqqii

2

02 NtnE

Hqiqi tnjtntnjtnEtnE 2

222 tnEtnEtnE qi

tnjtntnjtnEtnE qiqi 2

4

022 NtnEtnE qi

),(: nmrandnnMATLAB

2),(),(: sqrtnmrandninmrandnnMATLAB

24

Noise Envelope and Phase (1)

• Noise as a magnitude and phase ttf2costAtn ncn

nni cosAn nnq sinAn

• The magnitude is a Rayleigh distribution– Mean and moment

nR

2n

R

nnA Au

N2Aexp

NAAp

n

2NAE R

n

R2

n N2AE

25

Noise Envelope and Phase (2)

• Probability of An exceeding a

• Phase Distribution

• Noise Power

Rn N

aaAP 2exp2

2021

nn forp

2212

2cos

02

222

NNNtnE

ttfEtAEtnE

RR

ncn

nR

2n

R

nnA Au

N2Aexp

NAAp

n

26

Noise Characteristics

• The noise power does not change based on the representation, the center frequency, or due to mixing.

• The noise power will change when the bandwidth is further limited in some way!

27


Model of a CW communication system with noise: Figure 10.1-1

CW Communication with Noise

ttf2cosLtAtx cc

tnttf2cosLtAtv c

thtnttfLtAteD Rc

2cosPr

ttf2costAtx c

28


Figure 10.2-1

Model of receiver for CW modulation with noise

• Select the noise model that best fits the demodulation operation– Magnitude-phase– Quadrature

29

Synchronous Demodulation DSB (1)

• DSB

• Synchronous Detector: mix to baseband and LPF

tftntftntxAtv cqcic 2sin2cos

thtftftntftntxAty LPFccqcic 2cos2sin2cos

thtftntftntxAty LPFc

qc

ic

2

222sin2

222cos21

tntxAty iic 21

tntxtv c

30


• DSB Pre-D SNR (RF BPF bandwidth BT=2W)

22

21 tntxAEtyE iic

22 tntxEtvE c

TRc BNStnEtxEtvE 0222

21

2 000

2

Pr T

R

TT

R

T

xc

De BW

WNS

BW

BNS

BNSA

NS

• DSB Post-D SNR (BB LPF bandwidth BLPF=W)

LPFRLPFxc BNSBNSAtyE 0022 22

412

41

WNS

BNSA

NS R

LPF

xc

DPost 00

2

2

2

2xc

RSASwhere

WNSAwhere xc

0

2

2


• Pre-D vs. Post-D SNR

31

21

2 000

2

Pr T

R

TT

R

T

xc

De BW

WNS

BW

BNS

BNSA

NS

WNS

BNSA

NS R

LPF

xc

DPost 00

2

2

2

21

Pr

De

DPost

NSNS

• A factor of 2 improvement in the SNR!

32

Synchronous Demodulation AM (1)

• AM

• Synchronous Detector: mix to baseband, DC block and and LPF

tftntftntxAtv cqcic 2sin2cos1

thtftftntftntxAty ccqcic 2cos2sin2cos1

thtftntftntxAty cq

cic

2

222sin2

222cos211

tntxAty iic 21

tntxtv c

33


• AM Pre-D SNR (RF BPF bandwidth BT=2W) 2

c2 tntxEtvE

TXC

c BNSAtnEtxEtvE 02

2222 1

2

T0

X2

2c

T0

R

DePr BN

S12

A

BNS

NS

• AM Post-D SNR (BB LPF bandwidth BLPF=W)

22

21 tntxAEtyE iic

LPFXc BNSAtyE 0222 2

41

DeX

X

DeLPF

T

Xc

Xc

LPF

Xc

DPost NS

SS

NS

BB

SASA

BNSA

NS

Pr2

2

Pr22

22

0

22

12

1222

34

Synchronous Demodulation AM (3)• For = 1 and Sx = 0.5

WN

AWN

ABN

SA

BNS

NS cc

T

Xc

T

R

De

0

2

0

2

0

22

0Pr 835.01

4

12

DeDeDeX

X

cc

LPF

Xc

DPost

NS

NS

NS

SS

WNA

WNA

BNSA

NS

PrPrPr2

20

2

0

2

0

22

32

5.015.02

12

425.0

2

• SNR appears to decrease, but the “definition of the signal” changed from Pre-D to Post-D• The carrier was removed

35


• Defining

• Textbook use of for AM

WNSA xc

0

22

2

X

X

T

Xc

De SS

BN

SA

NS

2

2

0

22

Pr 211

2

WNSA

NS Xc

DPost 0

22

2

X

X

DeX

X

DPost SS

NS

SS

NS

2

2

Pr2

2

112

WN

SANS Xc

De

0

22

Pr 212

36

AM vs DSB Demod Comparison

WNSA

BNSA

NS xc

T

xc

De

0

2

0

2

Pr 42

WNSA

NS xc

DPost

0

2

2 WNSA

NS Xc

DPost

0

22

2

WN

SANS Xc

De

0

22

Pr 41

DeX

X

DPost NS

SS

NS

Pr2

2

12

DSB AM

DeDPost NS

NS

Pr

2

WNSA xc

DSB

0

2

2

WNSA Xc

AM

0

22

21

37

Synchronous AM Conclusions

• 67% or more of the Pre-D signal power comes from the carrier.– There is only 33% or less of the “signal” SNR for AM

as compared to DSB

• The Post-D SNRs for DSB and AM are the same• If the signal powers (SX) are identical, AM is

transmitting at least 3x the power of DSB to achieve the same output, Post-D SNR.

38

Envelope Demodulation AM

• AM Envelope Detection (non-coherent demod)

– where

tf2sintntf2costntx1tAtv cqcic

tntxtv c

ttf2costAtv vcv

2q2

icv tntntx1AtA

tntx1tA

tnarctant

ic

qv

A Phaser representation of the Signal + Noise

tf2jexptjexptARetv cvv

39

Envelope Demodulation AM

• Envelope Detection is the magnitude with a DC block

– Carrier dominate

tAEtAty vvD

The same as a coherent demodulator

tAEtntntx1Aty v2

q2

icD

tAEtntx1A

tn11tntx1Aty v2

1

2ic

2q

icD

22c nEA

tntxAty icD

40


Figure 10.3-1

Model for detection of exponential modulation plus noise

41

Demodulation of FM/PM

• Phase Isolation and Processing

– Where

• Apply a limiter to remove the AM and apply coherent downconversion to isolate the phase

ttf2costAttf2cosAtv ncncc

tntxtv c

ttcostAAttsintAarctant

nnc

nn

A Phaser representation of the Signal + Noise

2n

2c AEA

tttf2costtcostAAtv cnnc

42


• After Limiting (and bandpass filtering)

• Perform phase discrimination

– where

tttf2costtcostAAtv cnnc

tttf2cosAtpred cL

ttty

c

nn

nnc

nn

AtsintAarctan

ttcostAAttsintAarctant

2n

2c AEA

c

nn

AtsintAttsinttan

43


• After Extracting the Phase

• The noise PSD is

• Apply appropriate filters for PM or FM– PM: a low pass filter– FM: a derivative and low pass filter

2A

tntA

tsintAtty2

cc

nn

c

nnn A

tsintAty

Bfrect

AN

Bfrect

S2NfS 2

c

0

R

0yn

44

PM Post-D SNR

• PM Phase output

• SNR

• RF Input SNR

ttxty nPM

x2

PM0

Rx

2PM

0

Rx2

PM

R

0

x2

PM

PostD

SWN

SSWN

SS

SWNS

NS

TT0

R

eDPr BW

BNS

NS

45

FM Phase Diff and Filter

• FM Phase output

• Differentiate and LPF

• Noise PSD

tdttx2ty nFM

thdt

td21txth

dttdy

21 n

FM

Bfrect

S2Nf

Bfrect

S2N

21f2fS

R

02

R

02

2yn

46

FM Post-D SNR

• Noise PSD

• SNR

Bfrect

S2Nf

Bfrect

S2N

21f2fS

R

02

R

02

2yn

BNSS

B3

BNS3S

NS

NS

0

Rx2

2FM

30

Rx

2FM

FM

x2

FM

PostD

R

30

33

R

0B

B R

02FM S3

BN3B

3B

S2Ndf

S2NfN

x2

x

2FM

PostD

SD3SW

3NS

47


Including 12 dB deemphasis improvement of FM: Figure 10.4-1

Performance of CW Systems

48

Comparison

Type WBT

PostDNS γ thresh Complexity Comments

Baseband 1 1 Minor No Mod

AM 2 x

2x

2

S1S

20 Minor Envelope

DSB 2 1 Major Synch Demod

PM PMM2 x2PM S 10xb Moderate Phase, Const.

Amplitude

FM DM2 x2 SD3 10xb Moderate

Freq. Disc.,Const. Amplitude

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Chapter 10: Noise in Analog Modulation A. Bruce …homepages.wmich.edu/~bazuinb/ECE4600/Ch10_1.pdf6...

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