Noise in Linear Modulations - German University in Cairo...Dr. Ahmed El-Mahdy COMM 601: Modulation I Noise in Coherent SSB System (Cont.) • Then, the signal to noise ratio at the
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Dr. Ahmed El-Mahdy COMM 601: Modulation I
COMM 601: Modulation I
Lecture 8
Noise in Linear Modulations
Dr. Ahmed El-Mahdy COMM 601: Modulation I
White Noise• A white noise process has a flat PSD over a wide range
of frequencies:
is called the height of the power spectral
density of the noise . 2
oN
2)( o
w
NfS
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Ex: White Noise With zero mean
Time domain representation
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Narrow band white noise• Narrow band white noise : is a white noise passed through a
band pass filter which is generally used in the receiver.
• The power spectral density of Narrow band noise at the output of an ideal BPF with center frequency and bandwidth 2B is:
•
cf
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Time domain representation of narrow band noise
(1) In-phase and quadrature phase representation:
The in-phase and quadrature phase are low pass
Signals.
(2) Amplitude and phase representation:
)2sin()()2cos()()( tftntftntn cQcI
)(tnI )(tnQ
)(
)(tan)(,)()(
where
))(2cos()()(
122
tn
tnttntnr(t)
ttftrtn
I
Q
QI
c
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Power spectral density of the in-phase and quadrature phase components
where:
• Proof:
--------------------------
• The PSD of the in-phase and quadrature phase components is:
)()()()( cNcNNN ffSffSfSfSQI
)2sin()()2cos()()( tftntftntn cQcI
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in AM System Envelope Demodulation:
• The transmitted signal in an AM system has the form:
LPF
tdBPF
)(tnts i )(ty )(tVEnvelope Detector
)(ts
)2cos()](1[)( tftmkAts cac
)2sin()()2cos()]()](1[[
)()(
tftntftntmka
tntKsy(t)
cQcIac
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in AM System (cont.)
))(2cos()()( ttftRty ycy
)()]())(1([)(:where 22 tntntmkatR QIacy
)())(1(
)(tan)( 1
tntmka
tnt
Iac
Q
y
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in AM System (Cont.)
• The output of the envelope detector will be proportional to :
• The expression of can be simplified if we assume that the signal power is very large with respect to the noise power.
• We consider the strong signal case. The phasor diagram in this case is shown in the following figure. We expect that
to be much larger than most of the time. Hence we can approximate as
)(tRy
)(tRy
)(tnI
)(tnQ
)(tRy
y n
)(tRn
)(and)( tntn QI
)(tRy
a b c
d
)](1[ tmka ac
)()](1[
)(
tntmka
acoflengthadoflengthtR
Iac
y
)](1[ tmka ac
)()]())(1([)(:where 22 tntntmkatR QIacy
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in AM System (Cont.)• Hence the demodulator output is given by:
• The output from the LPF (assuming that the filter removes the dc component ) is given by:
• The output signal to noise ratio is given by:
)()](1[)( tntmkatd Iac
)()()( tntmkatV Iac
signal.message
theofpoweraveragetheis)()]([where 22 tmtmEPm
WN
PkaSNR
WN
tmEka
tnE
tmEkaSNR
o
maco
o
ac
I
aco
2
2
)]([
)]([
)]([
22
222
2
222
f
fHLPF
WW cf...
Dr. Ahmed El-Mahdy COMM 601: Modulation I
• SNRo in terms of average received power:
For AM signal, the received power is given by:
]1[2
]1[
:,2
]1[2
)]]([1[2
)]([
2
222
22
22
222
22
mao
mamaco
o
maco
mac
ac
r
PkWN
PkPkaSNR
asitwritecanweThenWN
PkaSNRSince
Pka
tmEka
tsKEP
ma
ma
o
ro
Pk
Pk
WN
PSNR
2
2
1
)2cos()](1[)( tftmkAts cac
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in AM System (Cont.)
• The output signal to noise ratio is given by:
Power
Efficiency of
AM
(less than one)
ma
ma
o
ro
Pk
Pk
WN
PSNR
2
2
1
The noise performance of AM is less than
DSB and SSB due to the wastage of
transmitted power
Dr. Ahmed El-Mahdy COMM 601: Modulation I
SNRo in AM for single tone modulating signal
The AM system must transmit 3 times as much as average power as DSB and SSB to achieve same performance.
WN
PSNRWhen
WN
PSNR
Ak
AP
o
ro
o
ro
m
am
m
3
1,1
2
,2
2
2
2
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in Coherent DSB System
CARRIER
BPF
f
fHLPF
WW cf...
W2
DSB Coherent Demodulation
tfc2cos2
LPF td )(2cos tntftmA icc )(2cos)( tntftmaty cc )(tV
W2
ccN
cc
c
m
mccT
cc
KAafS
tn
tntftma
Wf
tmP
PAtmAP
tftmAts
and)(
psdawithnoisenarrowbandis)(where
(**))()2cos()(
isfiltertheofoutputThe
.2bandwidthandfrequencycenterahasBPFThe
signal.message
theofpoweraveragetheis)(where
2
1)(
2
1
:ispowerdtransmitteaverageThe
2cos
2
222
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in Coherent DSB System (Cont.)Using the in-phase and quadrature phase components of the narrow
band noise:
We have,
)2sin()()2(cos)()( tftntftntn cQcI
)4sin()()4cos()]()([)]()([)2cos()(2)(
)2sin()()2(cos])()([)(
tftntftntmatntmatftytd
and
tftntftntmaty
cQcIcIcc
cQcIc
])()([)(
bygivenisoutputtheandfilterthebyremovedaretermsfrequencydoubleThe
tntmatV Ic
WNtn
Patma
oI
mcc
2:ispowernoiseoutputtheand)(iscomponentnoiseoutputThe
:ispowersignaloutputtheand)(iscomponentsignaloutputThe 2
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in Coherent DSB System (Cont.)• Then, the signal to noise ratio at the output is given by:
• From equation (**), we have
WN
PaSNR
o
mco
2
2
inputreceivertheatpoweraveragetheiswhere,2
1 2
rmcr PPaP
WN
PSNR
o
ro
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in Coherent SSB system
• Assume USB
)()( tnts i
tfc2cos2
LPF td
)(2sinˆ2cos)( tntftmatftmaty cccc
)(tVBPF
tftmAtftmAts cccc 2sinˆ2cos
W
)(2
have we*),*(*eq.from then,))(ˆ())((Since.and
)(psdawithnoisenarrowbandis)(where
*)*(*)()2sin()(ˆ)2cos()(
isfiltertheofoutputThe
.bandwidthandfrequencycenterahasBPFThe
222
mmc
rmcc
N
cccc
mc
PPa
PPtmEtmEKAa
fStn
tntftmatftma
ff
SSBN fS )(
cfcf Wfc Wfc
W2
DSBN fS )(
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in Coherent SSB system (Cont.)• Similar to the DSB case, we have
)4sin()]()(ˆ[
)4cos()]()([)]()([)2cos()(2)(
and
)2sin(])()(ˆ[)2(cos])()([)(
tftntma
tftntmatntmatftytd
tftntmatftntmaty
cQc
cIcIcc
cQccIc
])()([)(
bygivenisoutputtheandfilterthebyremovedaretermsfrequencydoubleThe
tntmatV Ic
WNtn
Patma
oI
mcc
:ispowernoiseoutputtheand)(iscomponentnoiseoutputThe
:ispowersignaloutputtheand)(iscomponentsignaloutputThe 2
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Noise in Coherent SSB System (Cont.)• Then, the signal to noise ratio at the output is given by:
• Using
WN
PaSNR
o
mco
2
inputreceivertheatpoweraveragetheiswhere,2
rmcr PPaP
WN
PSNR
o
ro
Which is similar to the Coherent DSB System
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Example• Compare the average transmitted power and channel bandwidth
requirements of DSB, SSB, and AM schemes for transmitting an audio signal with a bandwidth of 10 kHz with output signal to noise ratio of 40 dB. Assume that the channel introduces a 50 dB power loss and that the noise power spectral density at the receiver input is No/2= W/Hz. Assume for AM.
Solution
The Bandwidth requirements are:
For DSB and AM, the bandwidth= 2*10=20 kHz
For SSB, the bandwidth=10 kHz
1210
mo
ro
fN
PSNR :SSB&DSBFor
5.02 ma Pk
Dr. Ahmed El-Mahdy COMM 601: Modulation I
Watts60)10()10(6henceand
][)10(6)10)(2(3103
3
1
1SNR
detection, envelopeassumingAM,theFor
Watts20)10)(10)(2(
getwe),10(50islosspowerchanneltheSince
][)10)(2()10)(10)(2(*
havewe),10)(10)(2(,40with
54
484
0
2
2
o
54
5
448
412
AMT
mor
mo
r
ma
ma
mo
r
DSBT
moDSBr
moo
P
WfNSNRP
fN
P
Pk
Pk
fN
P
P
ordB
WfNSNRoP
fNdBSNR
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