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KetBingYongEE422SectionBProject
ErrorProbabilityResultforMQAMModulation
AbstractThisprojectisdesignedtostudytheSymbolErrorrateof2QAM,4QAMand16QAMreceivedsignal.EachSERwillbeplottedandcomparethegapbetweeneachSER.Infirstpart,wewillstudytheSERforeachmodulationwithAWGN.ThesecondpartisthemodulationtransmittroughtheRayleighfadingchannel.
SignalModelIngeneral,MQAMsignalistransmittedoveranonfadingadditivewhiteGaussiannoisechannelorafrequencynonselectiveRayleighfadingchannel.Thereceivedsignalisgivenby
wheren(t)isthenoiseelement.InAWGNchannel,n(t)isthewhiteGaussiannoiseandhasthechannelgain,h(t)=1.ButforRayleighchannel,~0,1isastandardnormalGaussianwithzeromeanandvarianceof1.TheplotbelowistheMQAMSimulinkmodelwithRayleighfadingchannel:
Figure1SimulinkmodelforRayleighfadingchannel
SymbolErrorRateofBPSK(2QAM)1.
Thebinarymodulationonlycorrespondtoonebit,thereforethesymbolandbiterrorratearethe
same.Considerthatthetransmittedsignals, cos2 totransmit0bitand
cos2 tosend1bit.Theprobabilityoferrorisgivenasbelow:
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2
InBPSK =2Awhere
Thereforethesymbolerrorrateisderivedas and
.
2.
TheanalyticalmethodusedintheBSPKSERcalculationisbasedontheformulaabove.TheQ
functioninMatlabisequivalenttoQ(x) = 0.5*erfc(x/sqrt(2)).
3. InBPSKstimulation,thereceivedsignalisconsideredas
.isa1sequenceandisaAWGNnoisewhichmeansboththerealandimaginaryvalueofnoisehasmaximummagnitudeof1.
4.
Thereceivedsignalisroundedto1sothattheoutputhassimilarformasinput.ThenthereceivedsignaliscomparedtothemessagesignaltofindthenumberoferroroccursandcomputedSERvalue.
5.
Aftertherunningwegettheplotforsymbolerrorrateagainstsignalnoiseratioforboththeoretical
calculationandstimulation.Bothcurvesareclosedtoeachother.Thisverifiedthevalidityofequation.
Figure2BERversusSNRofBPSK
0 1 2 3 4 5 6 7 8 9 1010-4
10-3
10-2
10-1
100
Average SNR per bit (dB)
Bit
erro
r rat
e
BPSK, theoryBPSK, simu
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SymbolErrorRateofQPSK(4QAM)1.
ThetransmittedsignalsofQPSKarecosandsin.Usingtheconstellation
point,weget 2.ThereforethesymbolerrorrateofQPSKisderivedas
2
where
.WeusedtheformulaabovetoplottheBERoutput.
2. InQPSKstimulation,thereceivedsignalisconsideredas .isa1j
sequenceandisaAWGNnoisewhichmeansboththerealandimaginaryvalueofnoisehasmaximummagnitudeof1.
3.
Thereceivedsignalisroundedto1jsothattheoutputhassimilarformasinput.ThenthereceivedsignaliscomparedtothemessagesignaltofindthenumberoferroroccursandcomputedSERvalue.
4.
TheplotforQPSKBERagainstSNRisfoundinbelow.Wenoticedthatboththecurvesforanalytical
calculationandstimulationissimilar.
Figure3SERversusSNRofQPSK
0 1 2 3 4 5 6 7 8 9 1010-3
10-2
10-1
100
Average SNR per bit (dB)
Bit
erro
r rat
e
QPSK, theoryQPSK, simu
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Symbo
1. The1thepr
where
Andt
And
2. For16
hasth
olErrorRa
6QAMmodurobabilityofc
e
heaveragee
.Therefor
6QAM,onlyheoutputBER
ateof16
ulationhasthcorrectdetec
.Thes
nergyofthes
re,thesymbo
analyticalforRversusSNR
QAM
Figur
heconstellatioctionisderive
symbolerror
signalsetis
olerrorratee
rmulaisrequiasbelow:
re416aryQAM
onabove,eacedas:
ratecanbed
.Hen
expressionis:
ired.Afterco
M
chsignalisse
derivedas:
nce,theenerg
mputingthe
eparatedby
gyperbit
equationabo
.Therefor
and
oveinMatlab
e,
b,we
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Figure5SERversusSNRof16QAM
GapsbetweenBPSK,QPSKand16QAM1.
ThesymbolerrorrateplotofBPSK,QPSKand16QAMarecombinedintoasingleplot:
0 1 2 3 4 5 6 7 8 9 10
10-0.6
10-0.5
10-0.4
10-0.3
10-0.2
10-0.1
Average SNR per bit (dB)
Sym
bol e
rror r
ate
QPSK, theory
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Figure6BERversusSNRforBPSK,QPSKand16QAM
2.
Fromtheplotabove,BPSKhasthelowestSERwhile16QAMhasthehighestSER.ThisimpliesthatthehigherMARYmayhavehighererrorrate.
3.
BPSKandQPSKhasconsistentgapdifferentof0.2.Whilethegapdifferentto16QAMisvaryingas
SNRisincreasing.Thegapdifferentof16QAMtoQPSKis0.6at0SNRand0.3at10SNR.Thegapdifferentof16QAMtoBPSKis0.8at0SNRand0.3at10SNR.
RayleighFadingChannel
1.
Ingeneral,theRayleighFadingchannelhasslow,flatfadingwithrespecttothesymbolperiod.Thesignaltonoiseratio(SNR)istakenasfixedoverthedurationofthedecisionintervalofonesymbol.Theaverageerrorprobabilitycanbecomputedbyintegratingoverthefadingdistribution.SotheSymbolErrorRate(SER)canbeachievedbyaveragingtheconditionalerrorprobabilitywithrespecttotherandomvariableasfollows
|
0 1 2 3 4 5 6 7 8 9 1010-4
10-3
10-2
10-1
100
Average SNR per bit (dB)
Sym
bol e
rror r
ate
BPSK, theoryBPSK, simuQPSK, theoryQPSK, simu16-QAM, theory
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where|istheprobabilityerroroferrorunderAWGN,and
istheprobabilityofSNR.In
RayleighFadingChannel, isanexponentialfunctionas
exp _andisanlocal
meanSNR.
SERforBPSKinRayleighFading
1. Inthebeginningoftheproject,weknowtheBERofBPSKis
or
.Fromtheequationabove,theBERforBPSKinRayleighfadingisabletobederived.
1 2 1
exp
121
1
2. ThetheoreticalvalueforBERversusSNRisasbelow:
Figure7BERversusSNRforBPSKinRayleighFading
3.
NoticethatBERofBPSKinRayleighFadingishigher,andthechangeofcurveisalmostconstantasSNRincrease.
0 2 4 6 8 10 12 14 16 18 2010-3
10-2
10-1
100
SNR (dB)
BE
R
Binary BPSK over Rayleigh Fading Channel
Theoretical BEREmpirical BER
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SERforQPSKinRayleighFading
1. TheBERofQPSKinAWGNis 2
=1 2
.|,theprobabilityofan
symbolerrorisderivedas1 1 .
TheSERofMQAMintheRayleighFadingchannelis 2 where 1
and
2
1 1/2.And 1/2and 1
.
21 1 2
2
tan 1 112
2.
TheplotbelowisthesymbolerrorrateversussignalnoiseratioofQPSKinRayleighFading
Figure8BERvs.SNRofQPSKinRayleighFading
3. NoticethatQPSKhashigherBERthanBPSKinRayleighFading.
ConclusionThisprojectgivesintuitionofsymbolerrorrateofdifferentMQAMboththeoreticalandexperimental
0 5 10 1510-4
10-3
10-2
10-1
100BER versus SNR of QPSK in Rayleigh Fading
Sym
bol E
rror R
ate
SNR
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Reference[1]AndreaGoldsmith,WirelessCommunications,1stEdition,CambridgeUniversityPress,2005.
[2]B.P.Lathi,ModernDigitalandAnalogCommunicationSystem,3rdEdition,2005.
[3]QuadratureAmplitudeModulation.April28,2008.
