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26.1
IntroductionSinusoidalcarrier: orwhereA
-amplitude(V);f0
-frequency(Hz);
0=2f0
-angularfrequency(rad/s);
-initialphase(rad).Basicmodulationsystems:ASKFSKPSK3Vectorrepresentationandvectordiagram46.2
2ASK(Binaryamplitudeshiftkeying(二进制信号)6.2.1BasicprincipleExpression:
where0
=2f0---istheangularfrequencyofcarrier
andA(t)isarectangularwaveform.
ModulationmethodsMultipliermethod:
envelopemaybenonrectangularSwitchingmethod:envelopemustberectangular5DemodulationmethodsEnvelopedetectionmethod
-carrierphaseinformationisnotused.Coherentdemodulationmethod-carrierphaseinformationisused.RemovenoiseRemovenoiseFlashDemo76.2.2PowerSpectralDensityAssumethegeneralexpressionofthe2ASKrandomsignalis
an
-binaryunipolarrandom
amplitude,itequalsto0or1
g(t)-symbolwaveform
T
-durationofthesymbol
thenthepowerspectraldensityofthe2ASKsignalcanbecalculatedas
式中,Ps(f)-powerspectraldensityofs(t)
PA(f)
-powerspectraldensityofA(t)
∴IfPA(f)hasbeenfound,thenPs(f)canbecalculatedbysubstitutingPA(f)intotheaboveequation.8FindPA(f):fromeq.(5.5-29):
wherefc
=1/T
G1(f)-frequencyspectrumofg1(t) G2(f)-frequencyspectrumofg2(t)
∵Now,g1(t)=0,∴Theaboveequationbecomes
whereG(f)=G2(f)
Andifg2(t)isrectangularpulse,thenSowehaveSo9Find
Ps(f):fromtheaboveequation:
When
P=½,theaboveequationbecomesAndTherefore,wehaveFinally,weobtain10CurvesofPA(f)
andPs(f)(b)CurveofPs(f)(b)CurveofPA(f)Aftermodulation,thebandwidthistwiceofbasebandsignal.116.2.3SymbolErrorProbabilityAssumethesignalafterbandpassfilteringequalstowhere
∵n(t)isanarrowbandGaussianprocess,wehaveSubstitutingtheabovetwoequationsintoy(t),weobtain:
.12
Coherentdemodulation
Thevoltagex(t)atthesampling&decisionpointis
wherenc(t)-Gaussianprocess
∴When1istransmitted,theprobabilityofx(t)
is
When0istransmitted,theprobabilityofx(t)is13Let
h
be
thedecisionthreshold,thentheprobabilityofthewrongdecisionof0as1
isequaltowhere
Theprobabilityofwrongdecisionof0as1is14WhenP(1)=P(0),thetotalsymbolerrorprobabilityis
Whenhequalstheoptimumthresholdh*,Whenr>>1,
15Envelopedetection
∵Theoutputistheenvelopeofitsinputvoltage
y(t),wehave
Ifthedecisiontheresholdequalsh,anditisprescribedthatwhenV>h,thedecisionisreceived1,andwhenV
h,thedecisionisreceived0,thenitssymbolerrorprobabilitycanbefiguredout.Whenthesignaltonoiseratioislarge,thesymbolerrorprobabilityis:16
【Example
6.1】Fora2ASKsignaltransmissionsystem,assumethesymbolrateisRB=4.8106Baud,theamplitudeofthereceivedsignalA=1mV,thesingle-sidepowerspectraldensityoftheGaussiannoisen0=210-15W/Hz.Find:1)theoptimumsymbolerrorprobabilitywhenenvelopedecisionisused;2)theoptimumsymbolerrorprobabilitywhencoherentdemodulationisused.
17Solution:Thebandwidthofthebasebandrectangularpulsecanbeselectedas1/THz.Thebandwidthofthe2ASK
signalshouldbetwiceofit,i.e.,2/THz.Henceaccordingtothesignalrate,theoptimumbandwidthofthebandpassfilterinthereceivershouldbeselectedas:B2/T=2RB=9.6106Hz
Thereforethemeanpoweroftheoutputnoiseofthebandpassfilteris:
Henceitsoutputsignaltonoiseratiois:
∴(1)thesymbolerrorprobabilityfortheenvelopedetectionis
(2)thesymbolerrorprobabilityforthecoherentdemodulationis186.3
2FSK 6.3.1BasicPrincipleExpression:GenerationmethodsFrequencymodulationmethod:
phasecontinuousSwitchingmethod:
phasediscontinuous2FSKproductionFlashDemo20ReceptionmethodsCoherentreceiving:Noncoherentreceiving:Envelopedetection:EnvelopeDetectionDemo22Zero-crossingpointdetectionZero-CrossingPointDetectionFlashDemo24
6.3.2
PowerSpectralDensity
2FSKsignalcanberegardedasthesuperpositionoftwo2ASKsignalswithdifferentfrequencies:
where∵Powerspectraldensityof2ASKsignalis ∴
thepowerspectraldensityof2FSKsignalshouldbethesumofthepowerspectraldensitiesoftwo2ASKsignals:
6.3.2
PowerSpectralDensity
∵
Substitutingitintotheaboveequation,thepowerspectraldensityof2FSKsignalisobtainedas:26
Whentheprobabilitiesoftransmitting1andtransmitting0areequal,P=½,theaboveequationcanbereducedas:WhereG(f)isthefrequencyspectrumofbasebandpulseand
Sowehave
27
Ascanbeseenfromtheaboveequation,theforgoing4termsarethecontinuousspectrumpartandthelatter4termsarethediscretespectrumpart.Curves:Bandwidth:28
6.3.3MinimumFrequencySpace
Inprinciple,iftwosignalsareorthogonal,thentheycanbeseparatedcompletely.
Fornoncoherentreception:let2FSKsignalbe
forsatisfactionoforthogonalcondition,require:
i.e.,require
29
6.3.3MinimumFrequencySpace
Theintegratedresultoftheaboveequationis
Assumethefirstandthirdtermsequalzero,thentheaboveequationcanbereducedtoSince1
and0
arearbitraryconstants,itmustsimultaneouslyhave
Hencewehave
So
30
itequalwhenm=1,itgivestheminimumfrequencyspace1/T.Forcoherentreceiving,letthentheequationcanbereducedto
Hence,itisonlyrequiredthati.e.,theminimumfrequencyspaceequals1/2T
.316.3.4
SymbolErrorProbabilityAssumetheoutputvoltagewaveformofthefilterisSymbolErrorProbabilityofCoherentDemodulation
Whenthesymbol1istransmitted,thetworeceivedvoltagesafterpassingthetwobandpassfiltersis:Theyaremultipliedbythelocalcarrier,andthenlow-passfiltered.Hence,weobtain
andNormalvariablewith(A,n2)Normalvariablewith(0,
n2)33Duringdecision,once
V1<V0,thesymbolerrorwillbeoccurred.Thereforethesymbolerrorprobabilityis
Let
(A+n1c-n0c)=z,then
z
isalsoanormalrandomvariable,anditsmeanequalsA,itsvarianceis
Hence,wehave
where
—signaltonoiseratio
∵Pe0
andPe1
areequal,thetotalsymbolerrorprobabilityis:34SymbolErrorProbabilityofEnvelopeDetection
Whenthesymbol1istransmitted,thetwovoltagesatthesampling&decisiondeviceinputrespectivelyare
and
whereV1(t)-envelopeofthesignalinthepathoffrequencyf1;(generalizedRayleighdistribution) V0(t)-envelopeofthesignalinthepathoffrequencyf0;(Rayleighdistribution)35weobtainwhere —signaltonoiseratio
When0istransmitted,thesituationoferroroccurringisthesameasabove.So,thetotalsymbolerrorprobabilityis36Comparisonofsymbolerrorprobabilitiesofcoherentdetectionandenvelopedetection:Thereisnobigdifferencebetweenthemundertheconditionoflargesignaltonoiseratio.Inpracticalapplication,envelopedetectionismoreoftenused.
Comparisonofsymbolerrorprobabilitiesof2FSK
and2ASKsignals:Envelopedetection2ASK:
3dBdiference2FSK:Coherentdetection2ASK:
3dBdifference2FSK:37【Example6.2】Assumethereisa2FSKtransmissionsystem,itstransmissionbandwidthis2,400Hz;theapparentcarrierfrequenciesofthe2FSKsignalarerespectivelyf0=980Hz
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