lecture 3-signal space, modulation

7/27/2019 Lecture 3-Signal Space, Modulation

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SignalSpaceandModulation


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Update

Haveconsidered

Losslesssourcecoding

Quantization

Sampling

Keyinsight:signal waveform vector

Wewill

now

consider

Signalspaceconcept

Modulation

Noisemodel

Modulationwithmemory

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Source

Encoder

Information

Source

Channel

EncoderModulator

ChannelNoise

Source

Decoder

Received

Information

Channel

DecoderDemodulator

Binaryinterface

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Update

Haveconsidered


Quantization

Sampling

Keyinsight:signalwaveform vector

Wewill

now

consider

Signalspaceconcept

Modulation

Noisemodel


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TheConceptofSignalSpace

Thesignalspace viewpointhasbeenoneofthefoundationsofmoderndigitalcommunicationtheory

sinceits

popularization

in

the

classic

text

of

Wozencraft andJacobs

Bychoosinganappropriatesetofaxisforoursignal

constellation,one

can:

Designmodulationtypeswhichhavedesirableproperties

Constructoptimalreceiversforagivenmodulation

technique Analyzetheperformanceofmodulationschemesusing

verygeneraltechniques

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Example1

Byinspection,

the

signals

can

be

expressed

in

terms

of

the

followingfunctions:

Theseareknownasbasisfunctions

Notethat Alsonotethateachofthesefunctionshaveunitenergy

WesaythattheyformanOrthonormalBasis

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)5.0(rect)(1 ttf )2/3(rect)(2 ttf

0*21

dttftf


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ConstellationDiagramofExample1

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Example2

Considerthefollowingorthonormalbasisfunctions:f1(t)andf2(t)(definedover[0,T)).

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)2cos(2)(1 tfT

tf c )2sin(2)(2 tfT

tf c

1as,0

4cos4

1

4sin2

12

2sin2

2cos2

0

0

0

0

*

21

Tf

tfTf

dttfT

dttfT

tfT

dttftf

c

T

c

c

T

c

T

cc

T

Thebasisfunctionsarethusorthogonal andtheyarealsonormalized


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ConstellationDiagramofExample2

Thesebasisfunctionsarequitecommonandcandescribevariousmodulationschemesbyproperly

selectingx(t)

and

y(t).

For

instance,

for

QPSK

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ThesignalconstellationisthesameasonSlide6


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SignalSpaceandBasisFunctions

Twoentirelydifferentsignalsetscanhavethesamegeometricrepresentation.

Theunderlyinggeometrywilldeterminetheperformanceandthereceiverstructureforasignalset.

Inthepreviousexamples,wewereabletoguessthe

correctbasis

functions.

Ingeneral,isthereanymethodwhichallowsustofindacompleteorthonormal basisforanarbitrarysignal

set? GramSchmidtOrthogonalization (GSO)Procedure

Wefirstconsiderfinitedimensionalvectorspacestogaininsights

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VectorSpaces

Avectorspaceisasetofelements ,calledvectors,alongwithasetofrulesforoperatingonboththesevectorsanda

set

of

scalars examples:,therealvectorspace,and,thecomplexvectorspace

Geometricinterpretationof

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VectorSpaces

Theaxiomsofavectorspacearelistedbelow

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VectorSpaces

Asetofvectors, , spans ifeveryvector isalinearcombinationof, ,

Avectorspace

isfinitedimensionalifthereexistsafinite

setofvectors, , thatspan,e.g.,. Asetofvectors, , islinearlyindependentif 0 impliesthateach is0 A

set

of

vectors

, , isdefinedtobeabasis for ifthesetislinearlyindependentandspans.Thenany canbeexpressedas

Thedimension ofafinitedimensionalvectorspaceisthe

numberofvectorsinanyofitsbasis

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InnerProductSpaces

Thevectorspaceaxiomscontainnoinherentnotionoflengthorangle,whichwillbeprovidedbyinnerproduct

Aninner

product

on

acomplex

vector

space

is

acomplex

valuedfunctionoftwovectors, ,denotedby , ,thatsatisfiesthefollowingaxioms

Avector

space

with

an

inner

product

satisfying

these

axioms

iscalledaninnerproductspace

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InnerProductSpaces

Forthevectorspace and theinnerproductsareusuallydefinedas

Thenorm orlength ofavector inaninnerproductspace

is

defined

as

, Twovectors and aredefinedtobeorthogonal( )if, =0

Asetofvectorsareorthonormaliftheyaremutually andallhave

unity

norm

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, and ,


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SignalSpaces

Generalizingthe

concept

of

vector

spaces

Thesetofallrealfiniteenergy signals ,denotedby,isarealvectorspace(checkit!)

Every

signal

has

an

Fourier

transform Asignalspaceisanysubspace

E.g.,thesetof signalsthataretimelimitedto[0,T] E.g.,thesetof signalswhoseFouriertransformsarenonzeroonlyin

Everysignalspace hasanorthonormalbasis ,

suchthatevery canbeexpressedas

where

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Innerproduct

Norm

Signalset isanorthogonal setif

If isanorthonormal set.Inthiscase,

Where

And we can write

SignalSpaces

Trytomakeananalogytofinitedimentional vectorspace

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KeyProperty

Givenanorthonormalbasis ,andtwofiniteenergysignals

Wehave

Orequivalently

i.e.,Innerproductsarepreservedinanorthonormalexpansion

Inparticular,

the

energy of

the

signal

can

be

calculated

as

Alltheinnerproduct/energycalculationcanbecarriedineitherrepresentation!

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OrthonormalExpansion

Ingeneral,givenasetofMsignals{si(t);i=1,2,.,M}definedoverR withfiniteEnergy. Thatis,

Then,wecanexpresseachofthesewaveformsasaweightedlinearcombinationoforthonormalsignals{k(t)}n

whereNM

Question:How

to

produce

an

orthonormal

basis?

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Projectionofasignalontoanother

Theprojectionofthesignalvontothesignalu isthesignalwthatsatisfiesbothofthefollowingconditions:

1. w=u2. vwisorthogonaltou

Itcan

be

shown

that

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u

v

w

Innerproduct

of

v

and

u

Normalizedu


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FiniteDimensionalProjection

LetSbeanndimensionalsubspaceofaninnerproductspaceV,andassumethat isanorthonormalbasisforS.Then

forany ,thereisauniqueprojectionthatsatisfies

Furthermore,theprojection isgivenby

Thevector isdenotedas ,calledtheperpendicularfromvtoS.So

andwecanget

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GramSchmidtOrthogonalization (GSO)

Procedure

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TheGramSchmidtProcedure

Remaining

basis

functions

are

found

by

removing

portions

of

signals

which

arecorrelatedtopreviousbasisfunctions,andnormalizingtheresult.


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Example

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OrthonormalSetsinL2

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Fourierseries

expansion

Sampling

functions


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InfiniteDimensionalProjection

Let beasequenceoforthonormalvectorsinL2,andletv beanarbitraryL2 vector

Thenthere

exists

aunique

L2 vector

u such

that

vu is

orthogonaltoeach ,m=1,2,n,and

Thisistheexactmeaningwhenwewrite

foran

infinite

dimensional

expansion

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Comparewith

finite

dimensionalcase


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Waveform DigitalSequence

ThesourceandchannelwaveformscanberepresentedasrealorcomplexL2 vectors

Given

an

orthonormal

basis

of

L2,

any

such

waveformcanberepresentedas

Forasinglesymboltransmission,finitedimensionalexpansion

Forcontinuoustransmission,ithelpstoconsiderinfinitedimensionalexpansion

Then

for

modulation

we

can

separately

design carrierwaveforms,i.e.,thebasisfunctions

signalconstellation,i.e.,thecoefficients

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Update

Haveconsidered


Quantization

Sampling


Wewill

now

consider

Signalspaceconcept

Modulation

Noisemodel


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DigitallyModulatedSignals

Designgoal: Generateasignalthatrepresentsthebinarydatastreamand

matchesthecharacteristicsofthechannel

Intransmission,blocks ofkbinarydigitalsaremappedintooneof waveforms

Whenthemappingisperformedwithoutany

constrainton

previous

waveforms

it

is

known

as

memoryless

Whenthemappingdependsonpreviouswaveformsthemodulatorhasmemory

Linearvs. Nonlinear

kM 2

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DigitalModulation

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Bitsto

signalsbinary

input

Signalsto

waveform

Basebandto

passband

Channel

Signalsto

bitsbinary

output

Waveform

tosignals

Passband to

baseband

Passband

waveform

Sequences

ofsignals

Baseband

waveform


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ChoiceofModulationPulse

APAMmodulatorisdeterminedbythesignalconstellationA ,thesignalintervalT,andtherealL2 modulationpulsep(t)

AstandardMPAMsignalconstellation

Thechoiceofp(t) ismorechallenging

p(t) shouldapproach

0rapidly

as

Itshouldbeessentiallybasebandlimitedtosomebandwidth

ItshouldbeorthonormaltoallitsshiftsbymultiplesofT

Theretrievalofthesignalsequenceshouldbesimple.Intheabsence

ofnoise,

should

be

uniquely

specified

by

the

received

waveform

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ComputingInnerProductswithaMatchedFilter

Wewanttocomputetheinnerproduct

Definethe

matched

filter

as

Then

Therefore

i.e.,we

can

compute

inner

product

by

passing

the

signal

throughamatchedfilter

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Orthonormality Conditions

Forasignal andatimeintervalT,thefollowingareequivalent

1. Thetime

shifts

are

orthonormal

2. Thecompositeresponse satisfies

and

3. The

Fourier

transform

satisfies

the

Nyquist

criterionforzerointersymbol interference,namely

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Nyquist Theorem

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RaisedcosineSpectrum

Objective:findg(t)thatareidealNyquistbutareapproximatelybandwidthlimitedandtimelimited

Nyquist

bandwidth

ActualbasebandbandwidthBb, Apracticalsolution:Raisedcosinefrequencyfunction

Bb exceedsWb byarelativelysmallamount

Itissmoothinorderforg(t)todecayquicklyintime,asymptotically

with1/t3

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Choosingp(t)andq(t)

Howtochoosep(t)andq(t)subjectto ?

Wechooseas ,so

ItiscalledassquarerootofNyquist

Ifp(t)isreal,then

q(t) iscalled

as

the

matched

filter

to

p(t)

Theactualbandwidthofg(t),p(t),q(t)arethesame

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BasebandtoPassband

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Double

sideband

amplitude

modulation Notbandwidthefficient

Halfofthebandisredundant(u(t)isreal)


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QuadratureAmplitudeModulation(QAM)

Baseband modulator(complexsignaluk)

Baseband demodulator

Samplethe

output

at

Tspaced

sample

times

Equivalentpassband expressions

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Differenceinenergy,

justascaling

Differentauthorsmayuse

differentscaling,Proakis

used1!

Payattentiontosuch

difference,thoughthe

performancewillbethe

same


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BasebandRepresentation

Analyticrepresentation(oranalyticsignal)

Itcanbeshownthat

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Basebandrepresentation

Itcanbeshownthat

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Sameaswhatwegoton

Slide40,justdifferent

waytointerpret


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Basebandrepresentationofconvolution

Basebandrepresentationoffiltering

Frequency

response

with

respect

to

the

bandwidth

W

around

thecarrierfrequencyfc is

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Passband vs.Baseband

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Passband Analytic Baseband

Timedomain

Frequency

domain

Bandwidth B B B/2

Energy


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QAM

Atpassband,QAMsignalcanalsobeshownasanorthonormalexpansion

Itcan

be

shown

that

areorthonormal

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Anotherwayoflookingat

theconversion

on

slide

40


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QAMModulationandDemodulation

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GeneralOrthonormalModulation

ConsideranMary signalsetofrealntuples

The

selected

signal

vector

is

modulated

into

a

signal

waveform

Bydoingthis,wemapsymbols0toM1intoasetofsignalwaveforms

Totransmitasequenceofsymbols,wechoosetheorthonormalwaveforms suchthat

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GeneralOrthonormalModulation

ForPAM

For

QAM

Withsuchorthonormalwaveforms,asequenceofsymbols,say eachdrawnfromthealphabet{0,,M1},could

bemappedintoasequenceofwaveforms

Thetransmitted

waveform

would

be

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PAMvs.QAM DoF

PAM RealsignalsgeneratedatTspacedintervals,transmittedina

basebandbandwidthalittlemorethan

Overan

asymptotically

long

interval

T0,

2WbT0 real

signals

can

be

transmitted

QAM ComplexsignalsgeneratedatTspacedintervals,transmittedina

passband bandwidth

a

little

more

than OveranasymptoticallylongintervalT0,WbT0 complexsignals,or2WbT0 realsignalscanbetransmitted

Weknowthatrealwaveformsoccupyingtimeinterval(T0/2,T0/2)andfrequencyinterval(W

0,W

0)hasabout2W

bT0degrees

of

freedom (DoF)

PAMandQAMusealltheDoFs availableinthegivenbands

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SignalConstellation

AnNdimensionalsignalconstellation isdenotedby

Itselements

is

called

signal

points

Basicparametersofasignalconstellation

ItsdimensionalN

Itssize

M

(number

of

signal

points)

Itsaverageenergy

Itsminimumdistance

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SignalConstellation

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PAM PSK

QAM

l d l l


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MultidimensionalSignals

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Orthogonal Biorthogonal

Simplex

SignalsfromBinaryCodes

U d


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Update

Haveconsidered


Quantization Sampling


We

will

now

considerSignalspaceconcept

Modulation

Noisemodel


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N i P


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NoiseProcess

Wehaveseenthatthesignalwaveformcanbeexpressedas

where isanorthonormalbasis

Fornoiseprocess,wewillconsiderrandomprocessesdefinedas

Z1,Z2,,isasequenceofrandomvariables

Foreacht,Z(t)isasumofrandomvariables

IfZ1,Z2,,areiid withmean0andvariance,thenthecovariancefunctionis

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Additi N i


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AdditiveNoise

Thechanneloutputis

X(t)and

Z(t)

are

random

processes

modeling

channel

input

andnoise

X(t)andZ(t)areindependent

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J i tl G i


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JointlyGaussian

EachofthefollowingarenecessaryandsufficientconditionsforarandomvectorZwithanonsingular

covariance

KZ to

be

jointly

Gaussian1. Z=AW,wherethecomponentsofWareiid normaland

2. Zhasthefollowingjointprobabilitydensity

3. AlllinearcombinationsofZareGaussianrandom

variables

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Gaussian Processes


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GaussianProcesses

isazeromeanGaussianprocessif,forallpositiveintegersnandallfinitesetsofepochst1,,tn,thesetof

randomvariables isa(zeromean)jointly

Gaussianset

of

random

variables.

AzeromeanGaussianprocessisspecifiedbyitscovariancefunction

Wewill

focus

on

the

Gaussian

process

defined

as

isacountablesetofrealorthonormalfunctions

Zk areiid

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Linear Functionals of WSS Processes


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LinearFunctionals ofWSSProcesses

ConsiderazeromeanWSSprocessZ(t),andarealL2 functiong(t).Forthelinearfunctional

Itcanbeshownthat

Similar,for itcanbeshownthat

Ifgm(t)areorthonormalandSZ(f)isconstant,then

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Spectraldensity

White Gaussian Noise


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WhiteGaussianNoise

ItisastationaryzeromeanGaussianprocessW(t)

Itscovariancefunctionis

Thiscanonlybeanapproximation,thisfunctionneedstobeverynarrowrelativetothevariationofallfunctionsofinterest

Thismeansthatthespectraldensityisapproximatedflat

Considerasetoforthonormalfunctions ,andlet

then

Whenthenoiseisrepresentedintermsofanyorthonormalexpansion,theresultingRVsareiid

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Additive Noise Model


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AdditiveNoiseModel

Considerthesignalwaveformgivenby

Thetransmittedpassband waveformis

Itcanbewrittenas

where

areorthonormal

Thewhitenoisewaveformcanbeexpressedas

Zk,j areiid Gaussian

isindependentofUk,j andZk,j

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Notehereweusea

differentscalingfactorto

assistthediscussion

AWGN Channel


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AWGNChannel

Thechanneloutputis

It

can

be

expressed

as

Afterdemodulation

Theresidualnoise isindependentof

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DifferentscalingfactorsduringmodulationmaychangethepowersofUkand

Zk,buttheratio,i.e.,theSNRwillbethesame.

Payattentiontothescaling! X(t)isinorthogonalororthonormalexpansion

ofUk

Different Parameters


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DifferentParameters

PAM

SymbolintervalT

Nominalbasebandbandwidth

W=1/2T

AverageenergypersymbolEs=P/2W

Noisevarianceperdimension

N0/2 SNR=Es/(N0/2)=P/N0W

Transmissionrate

Spectralefficiency

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QAM

SymbolintervalT

Nominalpaseband bandwidth

W=1/T

AverageenergypercomplexsymbolEs=P/W

Noisevariancepertworeal

dimension

N0

SNR=Es/N0 =P/N0W

Transmissionrate

Spectralefficiency

Update


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Update

Haveconsidered


Quantization Sampling


We

will

now

considerSignalspaceconcept

Modulation

Noisemodel


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Linear Modulation with Memory


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LinearModulationwithMemory

Usedtoprovidelowsidelobes inspectrum

NRZIisonesimplelinearexample

Importantnonlinearexamplesprovidespectrumshaping

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Non Linear Signals with Memory


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NonLinearSignalswithMemory

Forexample,forconventionalFSK

Abruptswitchingfromoneoscillatoroutputtoanotherresultsinlargespectralsidelobes

Thusit

requires

alarge

frequency

band

Itishoweverpossibletomakesignalswithmemoryhavetheminimumsseparationandyetremainorthogonalandalso

have

continuous

phase

Non-linear modulationwith memory

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Continuous phase FSK (CPFSK)


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ContinuousphaseFSK(CPFSK)

n

n nTtgItd )()(

t

d ddTfjT

Ets 0)(4exp

2)(

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Continuous phase FSK (CPFSK)


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ContinuousphaseFSK(CPFSK)

Re-expressed as

where

h is the modulation index

)(2

)(4);(

nTtqhI

dnTtgITfIt

nn

t

nnd

0);(2cos2

)( IttfT

Ets c

TtTtTt

t

tq

Ih

Tfh

n

kkn

d

2/102/

00

)(

2

1

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Continuous Phase Modulation


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ContinuousPhaseModulation

MoregeneralfromofCPFSKisCPMwhere

Whereq(t)issomenormalizedwaveformshape

Whenhk =h modulationindexisfixedforallsymbols.Whenitvariesknownasmultih

Ifg(t)=0

for

t>T

known

as

full

response.

If

the

pulse

lasts

longer

than

Tthen

known

aspartialresponse

n

k

kk nTtqhIIt )(2);(

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Continuous Phase Modulation


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ContinuousPhaseModulation

InfinitevarietyofCPMpossiblebychoosingdifferentg(t),handalphabetsize

Popularpulses

are

GMSK,

Rectangular

and

raised

cosine

GMSKwithBT=0.3isusedinGSM

kM 2

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CPMSignalSpaceDiagrams


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g p g

IngeneralCPMcannotbeeasilyrepresentedassignalspacediagrambecausephaseofcarrieris

timevariant

Insteadthebeginningandendpointsofthephasetrajectorycanbeeasilymarked

BinaryCPFSK

with

h=1/2

can

be

represented

in

signalspacehowever

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MinimumShiftKeying(MSK)


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MSK isaspecialformofbinaryCPFSKwhenh=1/2

Sincethis

produces

the

minimumfrequencyseparation (1/2T)fororthogonalsignalsover

lengthTwhile

also

continuousinphaseitisknownasminimumshiftkeying

Canalso

be

thought

of

as

4

phasePSKinwhichthepulseshapeishalfcycle

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PowerSpectrum


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Powerspectrumdensity(PSD) Helpsdeterminetherequiredtransmissionbandwidthofdifferent

modulationschemes

Comparebandwidth

efficiency

of

different

modulation

schemes

PSDobtainedfromFToftheautocorrelation

function))(*)(()()()( 2

txtxEdef

fj

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SpectralCharacteristics


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ThekeymotivationforperformingCPMistohavesignalswithlowsidelobes

Todeterminethespectralcharacteristicsof

memorylessmodulation

we

can

find

the

PSD

of

the

basictransmittedpulse

Forlinearlymodulatedsignals

Twoterms continuousandspectrallines

Spectrallines

disappear

when

the

information

symbol

meaniszero(thesymbolsequenceisuncorrelatedtoo)

m

iivvT

mf

T

mG

TfG

Tf

2

2

22

2

)()(

Pulse shape

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5360 74

PulseShaping


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Sincpulse

Raisedcosine

pulse

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CPMandCPFSK


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ForCPMandCPFSKthederivationis slightlymore

involved Fromthefigure:

Forh1,thespectrumismuch

broader

Ashapproaches1,thespectrabecomepeaked

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SpectralEfficiencyComparison


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Let

b

bT

R 1 = Bit rate where Tb=Bit duration

BPSK:1stnulloccursatfrequency=1/Tb=Rb.Then,

BBPSK,mulltonull=2Rb

QPSK: 1stnulloccursatfrequency=1/(2Tb)=Rb/2. Then,

Bmulltonull=Rb

MSK:1stnulloccursatfrequency=0.75/Tb=0.75Rb. Then,

BMSK,nulltonull=1.5Rb

Let

B99%

Bandwidth

which

contains

99%

of

total

power.

Then,

b

b

R

RB

8

2.1%99

(forMSK)

(forQPSK and OQPSK)

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B99%=8Rb

B99%=1.2Rb

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QuadratureModulation:Summary


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Fora90%powercontainment bandwidth, MSK andQPSK (sameforOQPSK)areaboutthesameandarethemostbandwidthefficient,withBPSK

requiringabouttwiceasmuchbandwidth.

For

a

99%

power

containment bandwidth,

MSK isbyfarthebest,followedbyQPSK,withBPSKbeingadistantthird.

Alltheseschemeshaveconstantenvelopes. Hence,theyareusedsothatnonlinearamplification isused(nonlinearamplifiersaremoreefficient

thanlinearones).

Inpractice,

filtering is

employed

and

this

results

in

envelope

deviation (which

prohibitstheuseofnonlinearamplification).

Byexperiment,ithasbeenfoundthatMSK sufferstheleast (intermsofenvelopedeviation),followedbyOQPSK,QPSK,andBPSK.

Thatiswhy,MSKforinstanceisusedinGSM andHYPERLAN (wirelessLAN)

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Summary


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Signalspaceconcept

Digitalmodulation

Noisemodel


Readingassignment

Ch57ofGallager

Ch3,9.2ofProakis

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lecture 3-signal space, modulation

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