通信原理英文版教学课件：class six-Mary SK

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.RemovenoiseRemovenoise76.2.2PowerSpectralDensityAssumethegeneralexpressionofthe2ASKrandomsignalis

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*,—signaltonoiseratioWhenr>>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:

phasediscontinuous20ReceptionmethodsCoherentreceiving:Noncoherentreceiving:Envelopedetection:22Zero-crossingpointdetection24

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

itequalswhenm=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:34SymbolErrorProbabilityofEnvelopeDetectionWhenthesymbol1istransmitted,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，f1=1580Hz;thesymbolrateisRB=300Baud,andthesignaltonoiseratioattheinputofthereceiveris6dB.Find: (1)Thebandwidthofthe2FSKsignal; (2)Theerrorsymbolprobabilityduringtheenvelopedetection; (3)Theerrorsymbolprobabilityduringthecoherentdemodulation.

38Solution:1.Thebandwidthofthesignal:2.Thesymbolerrorprobabilityofenvelopedetection:

Bandwidthofthebandpassfilter:B=2RB=600Hz

Thebandwidthratioattheinputandoutputofthebandpassfilter:2400/600=4

Thesignaltonoisepowerratioattheoutputofthebandpassfilter:r=4×4＝16

∴39

3.ThesymbolerrorprobabilityofcoherentdemodulationUsingtheerrorfunctiontableinAppendixB,weobtain

Iftheapproximationequation(6.3-27)isused,thenobtainTheresultsareapproximatelythesame.406.4

2PSK 6.4.1BasicPrincipleExpression:

where

or41Waveform

－“101”Discontinuousphase:

Fig.(a)andFig.(c) Thereisintegernumberofcarrierperiodsonasymbol.Continuousphase:Fig.(b)andFig.(d)

Thenumberofcarrierperiodsinasymbolisonehalfperiodlargerthanintegernumberofperiods.42Theaboveexampleillustrates:

Onlywhenasymbolcontainsintegernumberofcarrierperiods,(discontinuouscase),thephasejumpattheboundaryoftheadjacentsymbolsisthephasevariationproducedbymodulation.45 6.4.2PowerSpectralDensity

Ascanbeseenfromthe2PSKsignalequation:

The2PSKsignalcanberegardedasaspecial2ASKsignalwiththeamplitudesA

and–A.

∴Therandomsymbolsequenceofthe2PSKsignalcanalsobedescribedbytheexpressionofthe2ASKsignal:

where

Forthesimplificationoftheequationwithoutlostofthegenerality,wewillletA=1.46

Directlyfromtheequation:

where

for2PSKsignal,g(t)=-g(t)，G1(f)=-G2(f),andifP=½,

theaboveequationbecomesand

Thereisnodiscretefrequencycomponentintheaboveequation.－－can’tuseenvelopedetectionmethodbecausethereisnocarrierinfo.47

With

thefrequencyspectrumofarectangularpulse:

weobtainthefinalexpressionofthepowerspectraldensityofthe2PSK

signal:Comparisonofthepowerspectraldensitiesof2PSK

and2ASKsignals

Recallthatthepowerspectraldensityof2ASK

signalis:Thebandwidthsofbothareidentical.2PSKsignalhasnodiscretecomponent:

(f+f0)+(f-f0)48(a)Powerspectraldensityof2ASK

signal(b)Powerspectraldensityof2PSK

signal49 6.4.3SymbolErrorProbability

Thesampling&decisionvoltageis

Theprobabilityofwrongdeciding0as1is

Pe0=P(V<0/when0istransmitted)

Theprobabilityofwrongdeciding1as0is

Pe1=P(V>0/when1istransmitted)

SinceherePe0=Pe1

,∴Thetotalsymbolerrorprobabilityis

50

TheareaoftheleftshadowinthefigureisHence,thetotalsymbolerrorprobabilityis

or

Undercoherentdemodulationcondition,forobtainingthesamesignaltonoiseratio,therequiredpowerofthe2FSKsignalis3dBlargerthanthatofthe2PSKsignal;andthatofthe2ASKsignalis6dBlarger.516.5

2DPSK

6.5.1BasicprincipleExpression:

Assume

isthephasedifferenceofthecurrentsymbolandtheprevioussymbol,andlet

then,thesignalsymbolcanbeexpressedas

where0

＝2f0

---istheangularfrequencyofthecarrier

---phaseoftheprevioussymbol

Example:52Indirectgenerationof2DPSK

signalsFromviewpointofreceiver:can’tdistinguish2DPSK

and2PSKsignals.Forexample:ifthereceivedsymbolphaseis:

0

0

0for

2DPSK:

A=111001101（initialphase0）for2PSK:

B=

101110110（1）IfthesequencetobetransmittedisA,andconvertittothesequenceB,thenthelatterisusedtomodulatethecarrierby2PSK.Theresultisthesameas2DPSKmodulationdonedirectlybyA:Basebandsequence:

A=

111001101(Absolutecode)Sequenceafterconversion:B=(0)101110110(Relativecode)Phaseafter2PSK

modulation:

(0)00

0Conversionrule:1)

Absolutesymbol“1”changestherelativesymbol;

Absolutesymbol“0”doesn’tchangetherelativesymbol.2)53Conversionmethod:usingabistabletrigger

Blockdiagramofindirectmodulationof2DPSKsignal57Codeinverseconverter586.5.2PowerSpectralDensityThepowerspectraldensityof2DPSKsignalisthesameasthatof2PSKsignal.6.5.3SymbolErrorProbabilitySymbolerrorprobabilityofphasecomparisonmethodAssumethecontinuouslyreceivedtwosymbolsare00,then

Where

s0(t)－delayedwaveformoftheprevioussymbol

s1(t)－waveformofthecurrentreceivedsymbol59

Aftermultiplicationandlowpassfiltering,thesetwosymbolsbecome

Decisionrule:

IfV>0,thenthedecisionis0;

IfV<0,thenthedecisionis1.

Therefore,whenthecurrenttransmittingsymbolis0,theprobabilityofwrongreceivingequals

Byusingtheidenticalequation

theaboveequationcanberewrittenas

where60

－obeysgeneralizedRayleighdistribution;

－obeysRayleigh

distribution,

So

where.

So61SymbolErrorProbabilityofPolarityComparisonMethod

Ascanbeseenfromtheabovefigure,thefrontalpartofthedemodulationprocessisquitethesameasthatofthecoherentdemodulationofthe2PSK,soonlythesymbolerrorprobabilityintroducedbytheinversecodeconversion.

62【Example6.3】Assumetherateof1Mb/sisrequiredtotransmitdatabyusingthe2DPSKsystem,andthesymbolerrorprobabilitydoesnotexceed10-4,aswellasthesingle-sidepowerspectraldensityofthewhiteGaussiannoiseatthereceiverinputisn0

=

110-12W/Hz.Find:(1)

therequiredreceivedsignalpowerforthephasecomparisonmethod;(2)

therequiredreceivedsignalpowerforthepolaritycomparisonmethod.

Solution:

Nowthesymbolrateis1Mb/s.Thebandwidthoccupiedbythe2DPSKsignalisthesameasthatbytheASKsignal.Thus,thebandwidthofthereceivingbandpassfilteris

B

2/T=2106Hz

Theoutputnoisepowerofthebandpassfilteris63Whenphasecomparisonmethodisused:accordingtotherequirementtherequiredsignaltonoiseratioisandtherequiredsignalpowerisWhenpolaritycomparisonmethodisused,accordingtothesamerequirement,

i.e.,

Fromtheerrorfunctiontableweget:

Thereforetherequiredsignalpoweris646.6PerformanceComparisonofBinaryDigitalKeyingTransmissionSystemSymbolerrorprobabilitycurves656.7M-aryDigitalKeying

6.7.0signaltonoiseratio

r

－symbolenergytonoisesingle-side

spectraldensityratio

FortheM-arysystem,

onesymbolcontainskbitsofinformation:

k=log2

M

TheenergyEbofeachbitequalsE/k

,wherethesymbolenergyE

isequallydistributedineachbit.Consequently,

whererb

istheenergyperbittonoisesingle-sidepowerspectraldensityratio.

.666.7.1M-ASKMulti-levelunipolarNRZsignal

M-ASKsignal

（Fig.aFig.b）

Multi-levelbipolarNRZsignal

Suppressedcarrier

M-ASKsignal

（Fig.cFig.d）Theseareactually4ASK

signals:

Eachsymbolcontains2bits.67

68SymbolerrorprobabilityofsuppressedcarrierMASK

signal

whereM－magnitudeofthebase,orthenumberofamplitudes

r－signalmeanpowertonoisepowerratio

When

M=2,theaboveequationbecomes

696.7.2M-FSKBasicprincipleThesymbolsofM-FSKusethecarrieswithMdifferentfrequencies.70NoncoherentDemodulationBlockdiagram71 6.7.3M-PSKBasicprinciple:the

M-PSK

signalsymbolcanbeexpressedas wherek－themodulatedphase,itsvalueisdecidedbythevalueofthebasebandsymbol;

A－thesignalamplitude,aconstant

k=1,2,…,M

Let

A=1,thenthisequationcanbeexpandedas

where ThesymbolofM-PSKsignalcanberegardedasasignalcomposedoftwoorthogonalcomponentswithMkindsofamplituderespectively,i.e.,thesumoftwoM-ASKsignalwiththesamecarrierfrequency,henceitsbandwidthisthesameasthatofM-ASKsignal.

72QPSKCodingrule:A

modeandB

mode

GraycodeOnlyonebitdifferencebetweenadjacentkAdvantage:biterrorprobabilitysmallGrayBinary123400000010001100010000000100100010567801010111011001000100010101100111910111213141516110011101111110110011011101010001000100110101011110011011110111173GenerationmethodsThefirstmethod:multiplicationmethod

Binarysymbol1

Bipolarpulse+1

Binarysymbol0

Bipolarpulse-1BmodecodingNRZbipolarbasebandbinarysignal74Thesecondmethod:selectionmethod75Demodulationmethod－Coherentdemodulation76SymbolErrorProbability

Ifthephaseofthetransmittingsignal11is45,thenthedecisionthresholdsshouldbeat0

and90.

Assume:f()－probabilitydensityofthereceivedvector,thenthesymbolerrorprobabilityis:

Thecalculationresultoftheaboveequationis:776.7.4

M-DPSKBasicprinciple

Touse4-aryDPSK(QDPSK)

signalasexample,

Inthetable,k

isthephasechangerelativetotheprecedingadjacentsymbol.78Generationmethod TheinputsofmultipliersshouldbebinaryNRZbipolarrectangularpulses1and-1,thecorrespondingrelationshipis:

Binarysymbol0

＋1

Binarysymbol1

－1Amodecoding79Conversionre