Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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
Noise Equivalent Bandwidth
• 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
Noise Equivalent Bandwidth
• 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
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(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
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(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
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
thtnttfLtAteD Rc
2cosPr
ttf2costAtx c
28
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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
Synchronous Demodulation DSB (2)
• 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
Synchronous Demodulation DSB (3)
• 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
Synchronous Demodulation AM (2)
• 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
Synchronous Demodulation AM (4)
• 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
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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
Demodulation of FM/PM
• 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
Demodulation of FM/PM
• 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
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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