通信原理（英文版）第6章

Chapter6ElementaryDigitalModulationSystem6.1IntroductionSinusoidalcarrier:

orwhereA

－amplitude(V);f0

－frequency(Hz);

0=2f0

－angularfrequency(rad/s);

－initialphase

(rad).Basicmodulationsystems:ASKFSKPSKTTT“1”“1”“0”“1”“1”“0”T1Vectorrepresentationandvectordiagram26.2

2ASK

6.2.1BasicprincipleExpression: where0

＝2f0---istheangularfrequencyofcarrier

ModulationmethodsMultipliermethod:

envelopemaybenonrectangularSwitchingmethod:envelopeisrectangularMultipliercos0ts(t)A(t)cos0ts(t)A(t)3DemodulationmethodsEnvelopedetectionmethod－carrierphaseinformationisnotused.Coherentdemodulationmethod－carrierphaseinformationisused.Full-waverectifierBandpassfilterLow-passfilterSampling&decisionTimingpulses(t)A(t)EnvelopedetectorCoherentcarriercos0tMultiplicationcircuitBandpassfilterLow-passfilterSampling&decisionTimingpulses(t)A(t)46.2.2PowerSpectralDensity

Assumethegeneralexpressionofthe2ASKrandomsignalsequenceis wherean

－binaryunipolarrandom

amplitude

g(t)－symbolwaveform

T

－durationofthesymbol

thenthepowerspectraldensityofthe2ASKsignalcanbecalculatedas 式中，Ps(f)－powerspectraldensityofs(t)

PA(f)

－powerspectraldensityofA(t)

∴IfPA(f)hasbeenfound,then

Ps(f)canbecalculatedbysubstitutingPA(f)intotheaboveequation.5FindPA(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)

Now,thesymbolwaveformofthebasebandsignalisarectangularpulse,soitisknownfromFig.2.2.3andeq.(2.2-13),G(nfc)=0forallintegersofn

0.ThereforetheaboveequationbecomesSubstitutingtheeq.ofPA(f)intotheeq.ofPs(f),obtain:

6Find

Ps(f):fromtheaboveequation:

When

P=?,theaboveequationbecomes where

Therefore,wehave

Finally,weobtain7CurvesofPA(f)

and

Ps(f)f/fcPA(f)(a)CurveofPA(f)(b)CurveofPs(f)86.2.3SymbolErrorProbability

Assumethereceivedsignalandthenoisevoltageafterbandpassfilteringequal:

where

∵n(t)isanarrowbandGaussianprocess,wehave

Substitutingtheabovetwoequationsintotheequationofy(t),weobtain:orTheaboveequationisthereceivedvoltageafterbandpassfiltering.Itwillbeusedforcalculationofsymbolerrorprobability.9

Coherentdemodulation

Thevoltagex(t)atthesampling&decisionpointis

wherenc(t)－Gaussianprocess

∴When1istransmitted,theprobabilityofx(t)

is

When0istransmitted,theprobabilityofx(t)ish*Pe0p0(x)p1(x)Pe1hA10 Let

h

be

thedecisionthreshold,thentheprobabilityofthewrongdecisionof0as1isequalto where

Theprobabilityofwrongdecisionof0as1is

WhenP(1)=P(0),thetotalsymbolerrorprobabilityis

Whenhequalstheoptimumthresholdh*,

Whenr>>1,

h*Pe0p0(x)p1(x)Pe1hA11Envelopedetection

∵Theoutputistheenvelopeofitsinputvoltage

y(t),wehave

Ifthedecisiontheresholdequalsh,anditisprescribedthatwhenV>h,thedecisionisreceived1,andwhenV

h,thedecisionisreceived0,thenitssymbolerrorprobabilitycanbefiguredout.Whenthesignaltonoiseratioislarge,thesymbolerrorprobabilityis:12

【Example

6.1】Fora2ASKsignaltransmissionsystem,assumethesymbolrateisRB=4.8106Baud,theamplitudeofthereceivedsignalA=1mV,thesingle-sidepowerspectraldensityoftheGaussiannoisen0=210-15W/Hz.Find:1）theoptimumsymbolerrorprobabilitywhenenvelopedecisionisused;2）theoptimumsymbolerrorprobabilitywhencoherentdemodulationisused.

13Solution:Thebandwidthofthebasebandrectangularpulsecanbeselectedas1/THz.Thebandwidthofthe2ASKsignalshouldbetwiceofit,i.e.,2/THz.Henceaccordingtothesignalrate,theoptimumbandwidthofthebandpassfilterinthereceivershouldbeselectedas:B2/T=2RB=9.6106Hz

Thereforethemeanpoweroftheoutputnoiseofthebandpassfilteris:

Henceitsoutputsignaltonoiseratiois:

∴（1）thesymbolerrorprobabilityfortheenvelopedetectionis （2）thesymbolerrorprobabilityforthecoherentdemodulationis146.3

2FSK 6.3.1BasicPrincipleExpression:GenerationmethodsFrequencymodulationmethod:

phasecontinuousSwitchingmethod:

phasediscontinuousA(t)SwitchingcircuitFrequencysource0Frequencysource0s(t)

f1

f0FrequencymodulatorA(t)s(t)15ReceptionmethodsCoherentreceiving:Noncoherentreceiving:Envelopedetection:TimingpulseLowpassfilterLowpassfilterSampling&decisionOutputBandpassfiteringf0Bandpass

fiteringf1InputMultiplierMultipliercos0tcos1tV0(t)V1(t)y1(t)y0(t)Bandpassfilteringf0Bandpassfilteringf1EnvelopedetectionEnvelopedetectionSampling&decisionTimingpulseInputOutputV0(t)

V1(t)16Zero-crossingpointdetectionBandpass

filterAmplifier&limiterLow-passDifferen-tialRecti-fierPulsewideningabcedf17

6.3.2

PowerSpectralDensity

Thesymbolsequenceof2FSKsignalproducedbytheswitchingmethodcanberegardedasthesuperpositionoftwo2ASKsignalswithdifferentfrequencies:

where

∵Powerspectraldensityof2ASKsignalis ∴

thepowerspectraldensityof2FSKsignalshouldbethesumofthepowerspectraldensitiesoftwo2ASKsignals: ∵thepowerspectraldensityof2ASKsignalisknown:

Substitutingitintotheaboveequation,thepowerspectraldensityof2FSKsignalisobtainedas:18Whentheprobabilitiesoftransmitting1andtransmitting0areequal,P=?,theaboveequationcanbereducedas:WhereG(f)isthefrequencyspectrumofbasebandpulse:

andSubstitutingG(f)intotheaboveequation,weobtainthefinalexpressionofthepowerspectraldensityfothe2FSKsignal:19Ascanbeseenfromtheaboveequation,theforgoing4termsarethecontinuousspectrumpartandthelatter4termsarethediscretespectrumpart.Curves:Bandwidth:fsfs=(f0+f1)/2ff1+fcf0-fcf0f12fcf1fsff0fs=(f0+f1)/2fcf1+fcf0-fcfs=(f0+f1)/2f1+fcf0-fc20

6.3.3MinimumFrequencySpace

Inprinciple,iftwosignalsareorthogonal,thentheycanbeseparatedcompletely.Fornoncoherentreception:let2FSKsignalbe

forsatisfactionoforthogonalcondition,require:

i.e.,require

Theintegratedresultoftheaboveequationis

Assumethefirstandthirdtermsequalzero,thentheaboveequationcanbereducedto21Since1

and

0

arearbitraryconstants,itmustsimultaneouslyhave

andthentheaboveequationequalszero.

Hencewehave

and wheren

and

m

arebothintegers.

Forsimultaneoussatisfactionofthebothrequirements,itshouldbelet

i.e.,require

Therefore,whenm=1,itgivestheminimumfrequencyspace1/T.Forcoherentreceiving,letthentheequationcanbereducedto

Hence,itisonlyrequiredthati.e.,theminimumfrequencyspaceequals1/2T

.226.3.4

SymbolErrorProbability

AssumetheoutputvoltagewaveformofthefilterisSymbolErrorProbabilityofCoherentDemodulation Whenthesymbol1istransmitted,thetworeceivedvoltagesafterpassingthetwobandpassfiltersis:Theyaremultipliedbythelocalcarrier,andthenlow-passfiltered.Hence,weobtain

andTimingpulseLowpassfilterLowpassfilterSampling&decisionOutputBandpassfiteringf0Bandpass

fiteringf1InputMultiplierMultipliercos0tcos1tV0(t)V1(t)y1(t)y0(t)23

and

n1c(t)

andn0c(t)areGaussianprocess.Hence,theirsamplesV1

and

V0

areatthesamplinginstantarebothmormalrandomvariables.And,themeanofV1

is

A,thevarianceisn2;themeanofV0

is0,andthevarianceisalson2.

Duringdecision,once

V1<V0,thesymbolerrorwillbeoccurred.Thereforethesymbolerrorprobabilityis

Let

(A+n1c-n0c)=z,then

z

isalsoanormalrandomvariable,anditsmeanequalsA,itsvarianceis

Hence,wehave

where

∵Pe0

and

Pe1

areequal,thetotalsymbolerrorprobabilityis:24SymbolErrorProbabilityofEnvelopeDetection

Whenthesymbol1istransmitted,thetwovoltagesatthesampling&decisiondeviceinputrespectivelyare

and

whereV1(t)－envelopeofthesignalinthepathoffrequencyf1;(generalizedRayleighdistribution)

V0(t)－envelopeofthesignalinthepathoffrequencyf0;(Rayleighdistribution)Bandpassfilteringf0Bandpassfilteringf1EnvelopedetectionEnvelopedetectionSampling&decisionTimingpulseInputOutputV0(t)

V1(t)25Letandsubstitutingthemintotheaboveequation,afterreduction,wegetSubstitutingintotheaboveequation,weobtainwhere

—signaltonoiseratio

When0istransmitted,thesituationoferroroccurringisthesameasabove.So,thetotalsymbolerrorprobabilityis26Comparisonofsymbolerrorprobabilitiesofcoherentdetectionandenvelopedetection:Thereisnobigdifferencebetweenthemundertheconditionoflargesignaltonoiseratio.Inpracticalapplication,envelopedetectionismoreoftenused.

Comparisonofsymbolerrorprobabilitiesof2FSK

and

2ASKsignals:Envelopedetection2ASK：

3dB

diference2FSK：Coherentdetection2ASK：

3dBdifference2FSK：27【Example6.2】Assumethereisa2FSKtransmissionsystem,itstransmissionbandwidthis2,400Hz;theapparentcarrierfrequenciesofthe2FSKsignalarerespectivelyf0=980Hz，f1=1580Hz;thesymbolrateisRB=300Baud,andthesignaltonoiseratioattheinputofthereceiveris6dB.Find: (1)Thebandwidthofthe2FSKsignal; (2)Theerrorsymbolprobabilityduringtheenvelopedetection; (3)Theerrorsymbolprobabilityduringthecoherentdemodulation.

28Solution: 1.Thebandwidthofthesignal:

2.Thesymbolerrorprobabilityofenvelopedetection:

Bandwidthofthebandpass

filter:B=2RB=600Hz

Thebandwidthratioattheinputandoutputofthebandpassfilter:2400/600=4

Thesignaltonoisepowerratioattheoutputofthebandpassfilter:r=4×4＝16

∴29

3.ThesymbolerrorprobabilityofcoherentdemodulationUsingtheerrorfunctiontableinAppendixB,weobtain

Iftheapproximationequation(6.3-27)isused,thenobtainTheresultsareapproximatelythesame.306.4

2PSK 6.4.1BasicPrincipleExpression:

where

or31Waveform

－“101”Discontinuousphase:Fig.(a)andFig.(c) Thereisintegernumberofcarrierperiodsonasymbol.Continuousphase:Fig.(b)andFig.(d)

Thenumberofcarrierperiodsinasymbolisonehalfperiodlargerthanintegernumberofperiods.TTT“1”“1”“0”(c)(d)TTT(a)(b)“1”“0”“1”32Theaboveexampleillustratesthatthecontinuityoftheadjacentsymbolphasesshouldnotbeconfusedwiththeidentityoftheinitialphasesoftheadjacentsymbols.

Onlywhenasymbolcontainsintegernumberofcarrierperiods,thephasejumpattheboundaryoftheadjacentsymbolsisthephasevariationproducedbymodulation.33GenerationmethodsMultiplicationmethod

UsethebinarybasebandNRZbipolarrectangularpulseA(t)tomultiplythecarrierSelectionmethod

Usethebasebandsignaltocontrolaswitchingcircuit,andtheswitchingcircuitselectstheinputsignals.LocalcarrierA(t)s(t)MultiplierCarrierPhaseshifts(t)A(t)34DemodulationmethodCoherentreceivingmustbeused.Difficulties:(1)Thephaseofthelocalcarrierisdifficulttobedetermined.－thephaseambiguityofthefrequencydivider,andthechannelinstability

(2)Thesignalwithlongcontinuoussinusoidalwaveformmayappear,andhencethestartandthestopinstantsofasymbolcan’tbeidentifiedatthereceiver.

Toovercometheaboveshortcoming,theDPSKsystemcanbeused.

LocalcarrierextractorBandpassfilterLowpassfilterMultiplierSampling&decisionV(t)35

6.4.2PowerSpectralDensity

Ascanbeseenfromthe2PSKsignalequation:

The2PSKsignalcanberegardedasaspecial2ASKsignalwiththeamplitudesA

and

–A.

∴Therandomsymbolsequenceofthe2PSKsignalcanalsobedescribedbytheexpressionofthe2ASKsignal:

where

Forthesimplificationoftheequationwithoutlosingthegenerality,wewillletA=1.36

Directlyfromtheequation:

where

for2PSKsignal,g(t)=-g(t)，G1(f)=-G2(f),hencetheaboveequationbecomes

WhenP=?,theaboveequationbecomes

SubstitutingitintotheaboveequationofPs(f),weget

Thereisnodiscretefrequencycomponentintheaboveequation.－－can’textractcarrierfrequencydirectlyfromthereceivedsignalbyfilteringmethod.37

Substitutingthefrequencyspectrumofarectangularpulse:

intotheequation ,

wegetthefinalexpressionofthepowerspectraldensityofthe2PSKsignal

Comparisonofthepowerspectraldensitiesof2PSK

and

2ASKsignals

Thepowerspectraldensityof2ASK

signal:Thebandwidthsofbothareidentical.2PSKsignalhasnodiscretecomponent:

(f+f0)+(f-f0)38(a)Powerspectraldensityof2ASK

signal(b)Powerspectraldensityof2PSK

signal39Relationshipbetween2PSK

and

2ASK

signalwaveformsA2AA(a)2ASK(c)carrier(b)2PSK40

6.4.3SymbolErrorProbability

Thesampling&decisionvoltageis

Theprobabilityofwrongdeciding0as1is

Pe0=P(V<0/when0istransmitted)

Theprobabilityofwrongdeciding1as0is

Pe1=P(V<0/when1istransmitted)

SinceherePe0=Pe1

,∴Thetotalsymbolerrorprobabilityis

LocalcarrierextractorBandpassfilterLowpassfilterMultiplierSampling&decisionV(t)41

TheareaoftheleftshadowinthefigureisHence,thetotalsymbolerrorprobabilityis

or

Undercoherentdemodulationcondition,forobtainingthesamesignaltonoiseratio,therequiredpowerofthe2FSKsignalis3dBlargerthanthatofthe2PSKsignal;andthatofthe2ASKsignalis6dBlarger.0A-APe0Pe1V426.5

2DPSK

6.5.1BasicprincipleExpression:

Assume

isthephasedifferenceofthecurrentsymbolandtheprevioussymbol,andlet

then,thesignalsymbolcanbeexpressedas

where0

＝2f0

---istheangularfrequencyofthecarrier

---phaseoftheprevioussymbol

Example:000000

0

0

0(+)0Initialphase

00

0

00

0111001101111001101Basebandsignal43

VectordiagramAmode:thephasemayhavenojumppointforalongtime.Bmode:theremustbephasejumpbetweentheadjacentsymbols.

000

/2-/2ReferencephaseReferencephase(a) A

mode

(b)B

mode44Indirectgenerationof2DPSK

signalFromviewpointofreceiver:can’tdistinguish2DPSK

and

2PSKsignals.Forexample:ifthereceivedsymbolphaseis:

0

0

0

for

2DPSK:

A=111001101（initialphase0）for2PSK:

B=

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

Basebandsequence:

A=

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

modulation:

(0)000Conversionrule:

Absolutesymbol“1”changestherelativesymbol;

Absolutesymbol“0”doesn’tchangetherelativesymbol.

45Convesionmethod:usingabistabletrigger

Blockdiagramofindirectmodulationof2DPSKsignalCodeconverter(Bistabletrigger)AbsolutecodeRelativecodeA(t)CarrierPhaseshifts(t)Codeconverter46Demodulationof2DPSK

signalPhasecomparisonmethod:Disadvantage:requireveryhighdelayprecision;itisdifficulttoachievethat.Coherentdemodulationmethod:Thereceivedsignalisfirstcoherentlydemodulatedasanabsolutephaseshiftsignal;thenitisinverselyconvertedtoabsolutecode.MultiplierBandpassfilterLow-passfilterSampling&decisionV(t)DelayTs(t)A(t)s0(t)s1(t)LocalcarrierextractionMultiplierBandpassfilterLow-passfilterSampling&decisionInversecodeconversion47CodeinverseconverterCodeinverseconversionDifferential&rectifyingcbaPulsewidening(a)Blockdiagramc 1

1

1

0

0

1

1

0

1(Absolutecode)A

(0)

1

0

1

1

1

0

1

1

0(Relativecode)b(b)Waveform486.5.2PowerSpectralDensityThepowerspectraldensityof2DPSKsignalisthesameasthatof2PSKsignal.6.5.3SymbolErrorProbabilitySymbolerrorprobabilityofphasecomparisonmethodTwosignalstobecomparedbothcontainnoise.Assumethecontinuouslyreceivedtwosymbolsare00,then

Where

s0(t)－delayedwaveformoftheprevioussymbol

s1(t)－waveformofthecurrentreceivedsymbolMultiplierBandpassfilterLow-passfilterSampling&

decisionV(t)DelayTs(t)A(t)S0(t)s1(t)49

Aftermultiplicationandlowpassfiltering,thesetwosymbolsbecome

Decisionrule:

IfV>0,thenthedecisionis0,i.e.,thereceivingiscorrect;

IfV<0,thenthedecisionis1,i.e.,thereceivingiswrong.

Therefore,whenthecurrenttransmittingsymbolis0,theprobabilityofwrongreceivingequals

Byusingtheidenticalequation

theaboveequantioncanberewrittenas

orrewrittenas where50 －obeysgeneralizedRayleigh:distribution: －obeysRayleigh

distribution:

Substituting

f(R1)

and

f(R2)equationsinto

obtain

Theresultofintegrationis:

where51

Whenthetransmittingsymbolis1,thesymbolerrorprobabilityisthesame,hencewehave

∴Whentheprobabilitiesoftransmitting0andtransmitting1areequal,thetotalsymbolerrorprobabilityis52SymbolErrorProbabilityofPolarityComparisonMethod

Ascanbeseenfromtheabovefigure,thefrontalpartofthedemodulationprocessisquitethesameasthatofthecoherentdemodulationofthe2PSK,soonlythesymbolerrorprobabilityintroducedbytheinversecodeconversion.

LocalcarrierextractionMultiplierBandpassfilterLow-passfilterSampling&decisionInversecodeconversion53Symbolerrorprobabilityintroducedbyinversecodeconversion Whenoneandonlyoneofthetwoadjacentinputsymbolsofthecodeinverseconverterhasanerror,theoutputsymbolmaybeinerror. AssumethesymbolerrorprobabilityofthesignalattheinputofthecodeinverseconverterisPe;thentheprobabilityoftheprecedingsymboloftwosymbolsinerrorandthefollowingsymbolwithnoerrorequalsPe(1–Pe);andtheprobabilityoftheprecedingsymbolwithnoerrorandthefollowingsymbolinerrorisalsoequaltoPe(1–Pe). Thereforethesymbolerrorprobabilityoftheoutputsymbolis:

WhenPe

isverysmall,

WhenPe

isverylarge,i.e.,whenPe1/2,

54Finalsymbolerrorprobabilityequationofphasecomparisonmethod

Substitutingthesymbolerrorprobabilityequationof2PSKsignal

into

weobtain

or

When

Pe

isverysmall,wehave

55【Example6.3】Assumethetateof1Mb/sisrequiredtotransmitdatabyusingthe2DPSKsystem,andthesymbolerrorprobabilitydoesnotexceed10-4,aswellasthesingle-sidepowerspectraldensityofthewhiteGaussiannoiseatthereceiverinputisn0

=

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

therequiredreceivedsignalpowerforthephasecomparisonmethod;(2)

therequiredreceivedsignalpowerforthepolaritycomparisonmethod.

Solution:

Nowthesymbolrateis1MB.Thebandwidthoccupiedbythe2DPSKsignalisthesameasthatbytheASKsignal.Thus,thebandwidthofthereceivingbandpassfilteris

B

2/T=2106Hz

Theoutputnoisepowerofthebandpassfilteris56Whenphasecomparisonmethodisused:accordingtotherequirementtherequiredsignaltonoiseratioisandtherequiredsignalpowerisWhenpolaritycomparisonmethodisused,accordingtothesamerequirement,

i.e.,

Fromtheerrorfunctiontableweget:

Thereforetherequiredsignalpoweris576.6PerformanceComparisonofBinaryDigitalKeyingTransmissionSystemSymbolerrorprobabilitycurvesNoncoherentASKCoherentASKNoncoherentFSKCoherentFSKNoncoherentDPSKNoncoherentDPSK110-110-210-310-410-510-610-7PePSK-6-30369121518Signaltonoiseratior(dB)58SymbolerrorprobabilityequationsKeyingmodesErrorsymbolprobabilityequationsNoncoherent2ASK

eq.(6.2-46)Coherent2ASK

eq.(6.2-26)Noncoherent2FSK

eq.(6.3-34)Coherent2FSK

eq.(6.3-26)Noncoherent2DPSK

eq.(6.5-15)Coherent2DPSK

eq.(6.5-25)Coherent2PSK

eq.(6.4-13)596.7M-aryDigitalKeying

6.7.0signaltonoiseratio

r －symbolpowertonoisepowerratio

－symbolenergytonoisesingle-sidepower spectraldensityratio

FortheM-arysystem,

onesymbolcontainskbitsofinformation:

k=log2

M

TheenergyEbofeachbitequalsE/k

,wherethesymbolenergyE

isequallydistributedineachbit.Consequently,

whererb

istheenergyperbittonoisesingle-sidepowerspectraldensityratio.

WhenstudyingthesymbolerrorprobabilityfordifferentvaluesofM,itissuitabletouse

rb

astheunitforcomparison.606.7.1M-ASKMulti-levelunipolarNRZsignal

M-ASKsignal （Fig.aFig.b） Multi-levelbipolarNRZsignal

Suppressedcarrier

M-ASKsignal

（Fig.cFig.d）Theyare4ASK

signals:

Eachsymbolcontains2bits.(a)Basebandmulti-levelunipolarNRZsignal(b)M-ASKsignal0010110101011110000t0t0101101010111100000101101010111100000t00000t01011010101111(c)Basebandmulti-levelbipolarNRZsignal(d)SuppressedcarrierM-ASKsignal61BandwidthofM-ASKsignal

M-ASKsignalcanbeconsideredasthesuperpositionofseveral2ASKsignals.

∴Thebandwidthof themareidentical.ThefrequencybandwidthutilizationofM-ASKsignalexceedsNyquistcriterion:

forbasebandsignal

－2b/sHz

for2ASK

signal

－1b/sHzDisadvantageofM-ASK

signal:

Itisinfluencedbythechannelfading.0t0101101010111100000t01011010101111000001010t10101011110000001t1010101111000062SymbolerrorprobabilityofsuppressedcarrierMASK

signal

whereM－magnitudeofthebase,orthenumberofamplitudes

r－signalmeanpowertonoisepowerratio

When

M=2,theaboveequationbecomes

Itisthesymbolerrorprobabilityequationforthe2PSKsignalcoherentdemodulation.

110-110-210-310-410-510-6Per(dB)636.7.2M-FSKBasicprincipleThesymbolsofM-FSKusethecarrieswithMdifferentfrequencies.Assumef1

isthelowestcarrierfrequency,andfM

isthehighestcarrierfrequency,thenthebandwidthoftheM-FSKsignalisapproximatelyequaltofM-f1+f

,wheref

isthebandwidthofasinglesymbol,anditisdecidedbythesignaltransmissionrate.TTTTf3f1f2f464SymbolErrorProbabilityforNoncoherentDemodulationBlockdiagramOnlyoneoftheoutputsofthebandpassfiltersisthesignalplusnoise,andtheothersareonlynoises.Hence,theprobabilityofalltheenvelopesofthe(M-1)noiseslessthansomethresholdhequals

whereP(h)－probabilityoftheenvelopehoftheoutputnoiseofonefilter;itobeysRayleighdistribution.V0(t)Bandpassfilterf0Sampling&decisionEnvelopedetectionBandpassfilterfM－1EnvelopedetectionTimingpulseInputOutputVM－1(t)….….65FromtheRayleighdistributionequation:

whereN－envelopeofthefilteroutputnoise

n2

－powerofthefilteroutputnoiseTherewillbeerrordecisionoccurredifanyoneormorethanoneofthenoiseoutputenvelopesexceedthethresholdh,andtheprobabilityoftheerrordecisionequalsHowdoesitdecidethevalueofh

?

Theoutputvoltageofthebandpassfilterwhichhasthesymboloutputisthesumofthesignalandthenoise.ItsenvelopeobeysthegeneralizedRayleighdistribution:

wherex－

Envelopeofthesumofoutputsignal