Download - 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
Losslesssourcecoding
Quantization
Sampling
Keyinsight:signalwaveform vector
Wewill
now
consider
Signalspaceconcept
Modulation
Noisemodel
Modulationwithmemory
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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)).
ELEC5360 8
)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
Losslesssourcecoding
Quantization
Sampling
Keyinsight:signalwaveform vector
Wewill
now
consider
Signalspaceconcept
Modulation
Noisemodel
Modulationwithmemory
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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
Basebandrepresentation
Itcanbeshownthat
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Sameaswhatwegoton
Slide40,justdifferent
waytointerpret
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BasebandRepresentation
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
Losslesssourcecoding
Quantization Sampling
Keyinsight:signalwaveform vector
We
will
now
considerSignalspaceconcept
Modulation
Noisemodel
Modulationwithmemory
ELEC5360 53
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
ELEC5360 61
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
Losslesssourcecoding
Quantization Sampling
Keyinsight:signalwaveform vector
We
will
now
considerSignalspaceconcept
Modulation
Noisemodel
Modulationwithmemory
ELEC5360 63
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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5360 72
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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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
Modulationwithmemory
Readingassignment
Ch57ofGallager
Ch3,9.2ofProakis
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