lecture 5 synchronization

7/27/2019 Lecture 5 Synchronization

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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

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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


Synchronization

Carrierphase

estimation


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


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


Decisiondirected

loops



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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BlockdiagramforPAM

(DecisionfeedbackPLLDLPLL)

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BlockdiagramforQAM

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Update

Synchronization


Decisiondirected

loops



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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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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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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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


Decision

directed

loopsNondecisiondirectedloops


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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A h


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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

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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

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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


Decisiondirectedloops



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

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Example


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p

Showthattheestimatoronslide35isunbiased

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Summary


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y


Synchronization

Carrierphase

estimation


Jointestimation

Readingassignment

Ch5ofProakis

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lecture 5 synchronization

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