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CarrierandSymbolSynchronization
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Update
Haveconsidered
Digital(orthonormal)modulation
Detection
BUT,onlyforsimplifiedidealmodel
Wewill
now
consider
Noncoherentdetection
Synchronization
Carrierphase
estimation
Symboltimingestimation
Jointestimation
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Source
Encoder
Information
Source
Channel
Encoder Modulator
ChannelNoise
Source
Decoder
Received
Information
Channel
DecoderDemodulator
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TheCorrelation
Receiver Structure
I
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NeedNcorrelators
N:signaldimension
M:constellationsize
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TheCorrelation
Receiver Structure
II
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NeedMcorrelators
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NoncoherentDetection
Toimplementourdemodulator,atthereceiverweneed
eithertheorthonormalbasis
orthe
signal
set
Inmanycases,wedonothavesuchperfectinformation
Wirelesschannelwithrandomfading
Propagationdelay
Imperfectsynchronization
Solution
Noncoherentdetection,livewiththeuncertainty
Estimate
required
parameters,
synchronization
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TheOptimalDetectionwithUncertainty
ConsidertheAWGNchannelwithrandomparameter
Theoptimal
detection
(MAP)
rule
is
Wecanfollowourpreviousdevelopment
Decision
regions Errorprobability
ELEC5360 7
nsr m ,
dpsrpP
dpmrpP
mrpPm
mnm
m
m
)(maxarg
),|(maxarg
)|(maxarg
,
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NoncoherentDetectionwithRandomPhase
Considerthefollowingbasebandsignal
where
Fromthedetectionrule,wehave
Itcanbesimplifiedto
where isthemodifiedBesselfunction,whichisanincreasingfunctionofx
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EnvelopDetector
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NoncoherentreceiverforMFSKsignals
x(t)
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Noncoherentvs.CoherentDetection
ConsiderbinaryorthogonalFSKasanexample
Forcoherentdetection(weanalyzedthegeneralcasein
Lecturenote
4)
Fornoncoherentdetection(check4.5.3ofProakis)
As ,wehave
For ,thegapbetweencoherentandnoncoherentdetectionis
less
than
0.8
dB
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Update
Haveconsidered
Digital(orthonormal)modulation
Detection
BUT,onlyforsimplifiedidealmodel
Wewill
now
consider
Noncoherentdetection
Synchronization
Carrierphase
estimation
Symboltimingestimation
Jointestimation
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Synchronization
Anoftenneglectedsubjectofcommunicationsissynchronization ofthecarrierandsymbol
Inpractice,
there
is
propagation
delay,
carrier
offset
Inouroptimumstructuresitisassumedweknowthecarrierphaseandsymboltimingsatthereceiver
Theseare
needed
for
the
correlators and
integrators
at
the
receiver
Symbolsynchronizationisrequiredinallreceivers
Carrier
recovery
only
in
coherent
receivers Buthowdowedeterminethem?
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SignalModel
Oursignalmodelis
Wemustestimatebothphase (carrier)anddelay (timing)
Symboltiming
estimation:
Thesymbol
timings
are
dependent
on
thepropagationdelayandaccuracyoffewpercentofTisadequate
Carrierphaseestimation:ThecarrierphaseisdependentontheTxoscillatordriftaswellaspropagationdelay.Moreover,theaccuracyneededismuchhigherthanfewpercentofT(fc isnormallylarge)
Theyneedtobeseparatelyestimated
tfjjbcetzets
tntstr
2)()(Re2
)()()(
cf2
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Estimate
Thereforewecanwrite
Toestimatewecanusemaximumlikelihood(ML)or
maximumaposteriorprobability(MAP) criterion InMAP ismodeledasrandomandcharacterizedbya
prioripdf
In
ML is
deterministic
but
unknown
)();(
)(),;()(
tnts
tntstr
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Estimate
TheMAPestimateisthevaluethatmaximizes
TheMLestimate isthevaluethatmaximizes
Ifthereisnoprioriknowledgeof weassumethat p( )isuniform
MAPandMLarethenidentical
WewilluseMLhere
)|( rp
)(
)()|(
)|( r
r
r
p
pp
p
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OneshotEstimatevs.TrackingLoops
ThereceiverobservesoveranobservationintervalTo,andextractstheestimate
One
shot
estimate:
estimate
obtained
from
a
single
observation
interval
Inpracticeusuallydonecontinuouslyinatrackingloop
Trackingloop:Itcontinuouslyupdatestheestimate
Oneshot
estimates
yield
insight
for
tracking
loop
OneshotestimatesareusefulintheanalysisofMLestimation
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MLEstimate
SinceaddednoiseisGaussian,thejointpdf is
where
andTo isobservationinterval
N
n
nn
Nsr
p 12
2
2
)]([
exp2
1
)|( r
00
)();()()()(T
nn
T
nn dttftssdttftrr
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MLEstimate
Wecanthinkoftheargumentintheexponentintermsofacontinuousformofr(t)as
Nowmaximization
of
pdf is
equivalent
to
maximization
of
likelihoodfunction
0
20
)];([1exp)(T
dttsrN
0
2
012
2)];([
1
2
)]([lim
T
N
n
nn
N
dttsrN
sr
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CarrierRecoveryandSymbolSynchronization
BPSK
Carryrecoveryisusedtogeneratephaseestimate
Symbolsynchronizationisusedforthesamplerandpulsegenerator
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Mary PSKreceiver
Twocorrelators (ormatchedfilters)needed
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Mary PAM
Asinglecorrelator isrequired
AGC (AutomaticGainControl)isnowincludedtoeliminate
channelgain
variations
on
selection
of
thresholds
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Mary QAM
CombinesMPSKandMPAM
AGC isrequired
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Update
Synchronization
Carrierphaseestimation
Decisiondirected
loops
Nondecisiondirectedloops
Symboltimingestimation
Joint
estimation
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CarrierPhaseEstimation
PhaseerrorscausereductioninamplitudeofsignalspluscrosstalkbetweenIandQ
Why?
TheQAMandMPSKsignalcanbewrittenas
Demodulatedby
the
two
quadrature
carriers
The
output
QAM and PSK are moresensitive to phase error
)2sin()2cos()( tftBtftAts cc
)2sin(
)2cos()(
tftc
tftc
cq
ci
)
sin(2
1
)
cos(2
1
)(
)sin(2
1)cos(
2
1)(
tAtBty
tBtAty
Q
I
Crosstalkinterference
Power loss
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CarrierPhaseEstimation
Twobasicapproaches Multiplexaseparatepilotsignal
Derivephasefrommodulatedsignal(morepowerandspectrumleftforinformationsignal)
ML CarrierPhaseEstimation
Assumedelay
is
known
0 00
0
22
00
2
0
2
0
);(1
);()(2
)(1
exp
)];([1
exp)(
T TT
T
dttsN
dttstrN
trN
dttsrN
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CarrierPhase
Firstandthirdtermsareconstantsthatdonotaffectmaximization
Loglikelihoodfunctioncanalsobemaximized
0
);()(2exp)(0 T
dttstrN
C
0
);()(2
)(0 T
L dttstrN
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Unmodulated Carrier
Determinephaseofunmodulated carrier
Wewant
to
maximize
Maximizeby
finding
zero
of
derivative
implying
Orequivalently
0
)2cos()(2
)(0 T
cL dttftrN
A
0)2sin()(0
T c dttftr
0
0
2cos)(
2sin)(
tan 1
T
c
Tc
MLtdtftr
tdtftr
)()2cos()( tntfAtr c
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Unmodulated Carrier
Lastequationimpliesstraightforwardquadratureestimation
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Unmodulated Carrier
Otherconditionimpliesuseoflooptoextracttheestimateasshown
TheloopfilterisanintegratorwithbandwidthproportionaltoreciprocalofintervalTo
TheVCOisassumedtohaveinstantaneousphase
VCO
0
()T
dt
)2sin( MLctf
)4sin(2
1)sin(2
1
)2sin()2cos()(
tf
tftfte
c
cc
)()2cos()( tntfAtr c
t
c
dttvKt
ttf
)()(
),(2
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PhaseLockedLoop(PLL)
PLLplayskeyroleinformingthelooprequired
AssumeinputtoPLLis
OutputofVCOis
Productis
LoopFilter
VCOOutput
)2cos( tfc
)2sin( tfc
)4sin(2
1)sin(
2
1
)2sin()2cos()(
tf
tftfte
c
cc
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PhaseLockedLoop(PLL)
Loopfilterleavesonlythelowfrequencyphaseterm
TheVCOhascontrolvoltagev(t) andproducesinstantaneous
phase ThereforecanthinkofPLLas
BecauseoftheintegrationatVCOv(t) willtendtozerooncethecorrectphaseisfound
t
cc dvKtfttf )(2)(2
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Example The probability of error for binary PSK demodulation and detection
when there is a carrier phase error is
Suppose that the phase error from the PLL is modeled as a zero-meanGaussian random variable with variance . Determine theexpression for the average probability of error (in integral form).
e
e
be
NQP
2
0
2 cos2
)(
2
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Update
Synchronization
Carrierphaseestimation
Decisiondirected
loops
Nondecisiondirectedloops
Symboltimingestimation
Jointestimation
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DecisionDirectedLoops
Sofarwehavenotconsideredestimationwhenthesignalsaremodulatedwithinformation onlyconsideredthe
unmodulated carrier
Modulationwillcausethephaseofcarriertochange
Twoapproachestohandlethis:
Decisiondirected usesthedetectedinformationsequenceatthe
receiverin
the
carrier
estimation
problem
Nondecisiondirected doesnotuseanyinformationaboutthedetectedinformationintheestimationproblem averageitout
Considertheequivalentlowpassmodulatedsignal
)()()()()( tzetstznTtgIetr j
n
bn
j
b
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LikelihoodFunctions
LogLikelihoodfunction
Differentiating
Decision
Directed
dtnTtgtryyIN
e
KTTedttstrN
Tn
nT
bn
K
n
nn
j
o
j
T
bbL
)()(1
Re
n timeobservatioassume)()(1
Re)(
*
)1(1
0
*
0
*
0 0
1
0
*1
0
*1 ReImtanK
nnn
K
nnnML yIyI
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DecisionDirectedEstimate
BlockdiagramforPAM
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DecisionDirectedEstimate
BlockdiagramforPAM
(DecisionfeedbackPLLDLPLL)
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DecisionDirectedEstimate
BlockdiagramforQAM
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Update
Synchronization
Carrierphaseestimation
Decisiondirected
loops
Nondecisiondirectedloops
Symboltimingestimation
Jointestimation
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NonDecisionDirectedLoops
Treatdataasrandomvariablesandsimplyaveragethelikelihoodfunctionovertheser.v.
Key
idea
is
to
remove
the
information
bearing
component
to
obtainunmodulated carrier
Assumeeitherpdf isknownorapproximate
ConsiderBPSKwith+/Aamplitudes
Likelihoodfunctionconditionalonsignofamplitude
Averageoverbothsignswithequalprobability
T
dttstr
N
C );()(2
exp)(0
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NonDecisionDirectedLoops
Averageoverbothamplitudesigns
DifferentiatetoobtainMLestimate nonlinear soneedapproximations
ForMary thesituationisevenworse assumesymbolsarecontinuousr.v. assumeGaussian
T
TT
dtfttrN
dtfttr
N
dtfttr
N
00
0000
)2cos()(2
cosh
)2cos()(2
exp
2
1)2cos()(
2exp
2
1)(
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NonDecisionDirectedLoops
ConsiderBPSKbutnowweassumetheamplitudeisGaussianinsteadof+/A
Then
LogLikelihoodfunctionisquadraticunderGaussiansignals
Ifcrosscorrelationbetweensignalandreceivedsignalsis
smallthen
it
is
also
good
approximation
for
previous
case
too
(cosh x=x2 whenxissmall)
2/2
2
1
)(
A
eAP
2
00
)2cos()(2
exp)(
T
dtfttrN
C
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NonDecisionDirectedLoops
AverageoverKsymbols
Takelogsanddifferentiateweget
1
0
2)1(
0
)2cos()(2exp)( K
n
Tn
nT
dtfttrN
C
Tn
nT
K
n
Tn
nT
dtfttrdtfttr)1(1
0
)1(
0)2sin()()2cos()(
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NonDecisionDirectedPLL
Suggeststrackingloopagain
Removethe sign
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Example
BasedontheMLcriterion,determineacarrierphaseestimationmethodforbinaryonoffkeyingmodulation
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SquaringLoop
Moreintuitiveapproach
WidelyusedinpracticeforDSBmodulationsuchasPAM
Averageofs(t) willbezeroifPAMsymbolsallsymmetricalaboutzero
Thereforesquarethesignalinstead
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SquaringLoop
Aftersquare,thereispowerat2fc Butsigninformationremoved
Thefrequency
component
at
2fc is
use
to
drive
aPLL
Squaringoperationleadstoincreaseinnoise
Thereisaphaseambiguityof180o
Differentialencoderisneeded
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CostasLoop
BlockDiagram
Removethe sign
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MultiplePhaseSignals
ThesemethodscanbegeneralizedtoMphasemodulation
Keyistoremovetheinformationbearingcomponenttoobtainunmodulated carrier
Generalizedsquareloop
GeneralizedCostasloop
Multiplysignal
by
M phase
shiftedcarriers
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Comparisons
DFPLLdifferfromCostasandsquaringlooponlyinthewaythemodulationisremoved
DFPLLis
superior
in
performance
since
noise
effect
is
notassevere
ResultsshowthatDFPLLphaseerrorshavevariance
410
times
smaller
than
Costas
loop
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Update
Synchronization
Carrierphaseestimation
Decision
directed
loopsNondecisiondirectedloops
Symboltimingestimation
Jointestimation
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S b l Ti i E i i
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SymbolTimingEstimation
Outputofdemod mustbesampledperiodicallyatthesymbolrateatprecisesamplinginstants
where isthepropagationdelay Requireclocksignalatreceiveranditsgenerationis
knownas
symbol
synchronization
or
timing
recovery
Mustknowfrequency1/Tandsamplinginstant ortimingphase
Severalapproaches
mTtm
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Approaches
1. BothRxandTx synchronizedtomasterclock
receiverthenmustonlyestimatetimingphase
Goodfor
very
low
frequency
(VLF)
band
2. Simultaneouslytransmitclockfrequencyalong
withinformationsignal
WastesBW
and
power
GoodwhenhaveMultipleusersasoverheadsharedandsmallonperuserbasis
3. Clock
signal
extracted
from
the
modulated
signal
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ML Ti i E ti ti
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MLTimingEstimation
AgaincanuseML
Decisiondirectedandnondecisiondirected
InDecisiondirected
)();(
)();()(
nn nTtgIts
tntstr
nnn
Tn
n
T
L
yIC
dtnTtgtrIC
dttstrC
)(
)()(
);()()(
0
0
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ML E ti t
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MLEstimate
Differentiate
AgainatrackingloopwithVCC Voltagecontrolledclock
0)( n
nn yd
dI
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Non Decision Directed
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NonDecisionDirected
Useaveragingagain ForBPSK
Since forsmallx
FormultilevelwecanuseGaussianApprox again
Differentiatingwe
get
n nL Cy )(coshln)(
2
2
1coshln xx
n
nL yC )(21)( 22
n n
nnn
d
dyyy
d
d0
)()(2)(2
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Tracking Loop
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TrackingLoop
Loops
ELEC5360 58
Early late gate Synchronizes
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EarlylategateSynchronizes
Usessymmetrypropertiesofsignal Considerthefollowingrectangularpulse
Propersamplingtimeisatmaximum
Butnoise
makes
this
difficult
Butcorrelationisevenfunctionsoinsteadofsearchingforpeaktaketwovalueseithersideofpeakandaverageovertime
Thereforesuggesttheearlylategatesynchronizer
early optimum late
samples
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Rectangularpulse Matchedfilteroutput
Early Late Gate Synchronizers
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EarlyLateGateSynchronizers
BlockDiagram
ELEC5360 60
Differencebetween
correlator outputs
EarlyLate Gate Synchronizers
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EarlyLateGateSynchronizers
ApproximatesMLestimator
Substitutingweget
Thismathematicalexpressiondescribesthefunctionsperformedbytheearlylategatesymbolsynchronizeronthepreviousslide
2
)()()(
d
d
n TT
n
nn
dtnTtgtrdtnTtgtrC
yyC
d
d
222
222
00
)()()()(4
)()(
4
)(
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Update
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Update
Synchronization
Carrierphaseestimation
Decisiondirectedloops
Nondecisiondirectedloops
Symboltimingestimation
Jointestimation
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Joint Estimation
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JointEstimation
Jointestimationusuallyprovidesestimatethatisasgoodorbetterthanseparateestimation
Decisiondirectedjoint
trackingloopinQAM/PSK
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Estimation Performance
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EstimationPerformance
Performancedefinedintermsofbias andvariance
Thebiasofanestimateisdefinedas
Theestimateisunbiased ifbias=0
Varianceis
defined
by
)( xEbias
222 )]([)]([ xExE
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Estimation Performance
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EstimationPerformance
Varianceisdifficulttocompute
ButcanuseaCramerRao lowerbound
Lowerboundisveryusefulasitactsasabenchmark
Whenunbiased,numeratorisunityandweknowp(x|) isproportionaltologlikelihoodfunction
2
2
)|(ln
)(
)(
xpE
xE
xE
)(ln
1
)(ln
1)(
2
22
22
EE
xE
2 different forms of
the C-R boundELEC5360 65
Performance
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Performance
Anyestimatorthatisunbiasedandapproacheslowerboundisknownasanefficientestimate
Efficientestimates
are
rare
AwellknownresultisthatanyMLestimateisasymptotically(withunlimitedobservations)unbiased
andefficient
Varianceusually
inversely
proportional
to
SNR
or
signal
powermultipliedbyobservationinterval
Decisiondirectedsystemsgenerallyobtainthelower
bound
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Performance
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Performance
Symboltiming
ELEC5360 67
Example
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p
Showthattheestimatoronslide35isunbiased
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Summary
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y
Noncoherentdetection
Synchronization
Carrierphase
estimation
Symboltimingestimation
Jointestimation
Readingassignment
Ch5ofProakis
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