Lecture 4-Detection, Performance Analysis

7/27/2019 Lecture 4-Detection, Performance Analysis

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Detection


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Update

Haveconsidered

Signalspaceconcept

Modulation

Noisemodel

Wewill

now

consider

Optimaldetection

Errorprobabilityanalysis

Channelcapacity

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Source

Encoder

Information

Source

Channel

Encoder Modulator

ChannelNoise

Source

Decoder

Received

Information

Channel

DecoderDemodulator

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Update

Haveconsidered

Signalspaceconcept

Modulation

Noisemodel

Wewill

now

consider

Optimaldetection


Channelcapacity

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StatisticalDecision

Theory

Demodulationanddecodingofsignalsindigital

communications

is

directly

related

to

Statistical

DetectionTheory.

Givenafinitesetofpossiblehypotheses and

observations,

we

want

to

make

the

best

possible

decision(accordingtosomeperformancecriterion)

aboutwhichhypothesisistrue.

In

digital

Communications,

hypotheses are

the

possiblemessagesandobservations aretheoutput

ofachannel.

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DetectionTheory

HaveM possiblehypotheses Hi (signalai )withPi =

P(mi);

i=1,2,

,

M Theobservable issomecollectionofNrealvalues,

denotedby r=(r1,r2,,rN)withp(r|ai)

Goal:

Find

the

best

decision

making

algorithm

in

the

sense

ofminimizingtheprobabilityofdecisionerror.

Message DecisionChannel ia

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ObservationSpace

Ingeneral,rcanberegardedasapointinsome

observationspace

Eachhypothesis Hi isassociatedwithadecisionregion Di ThedecisionwillbeinfavorofH

i(H

iistrue)ifr isinD

i.

D1D

2

D

3

D4

DecisionSpace(M=4)

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MinimumProbabilityofErrorDecisionRule

Pe Computations(decisionis ):

MinimizePemaximize1Pe

OptimumDecision

Rule:

drrprP

drrprPP

PP

sentisa

decisioncorrectdecisioncorrect

decisioncorrect1)error(

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MaximumaPosteriori Probability

(MAP)Detector

ByBayesrule:

Detectors

based

on

LHS

are

known

as

maximum

a

posterioriorMAP sincehypothesisafterobservationr

Ifwe

need

to

know

something

about

a before

observation(RHS)thenweneedpriorknowledge

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BinaryDetection

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MAPdetector

Equivalently

Sufficientstatistic

Any

function

of

the

observation

from

which

the

likelihood

ratio

can

be

calculated

e.g.,thelikelihoodratio oranyonetoonefunctionof

Aparticularlyimportantoneisloglikelihoodratio(LLR)

Likelihood

ratio(LR)

Threshold


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ProbabilityofDetectionError

Decisionregion:

Assumethe

transmitted

signal

is

a0 or

a1,

the

probability

of

erroris

Overallprobabilityoferroris

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AdditiveGaussianNoise

Consider2PAMmodulation

Noise

Sowe

have

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FromBinarytoTheGeneralCase

Frombinarydetection,wehavelearned

MAP,Thresholddetection

Sufficient

statistics,

LLR Decisionregion errorprobabilityanalysis

ErrorprobabilitywithAWGN,

Forgeneraldetectionproblem

MAP,threshold

detection

LLR

Decisionregion

Pairwiseerrorprobability Unionbound

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Mary Detection

ConsiderasingleMary modulationsymbol

Directextensionfromthebinarydetection

TheMAP

rule

Wecanalsoconsiderthelikelihoodratio

Thenwecandeterminethedecisionregionandcalculatethe

errorprobability

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DetectionwithArbitraryModulation

Recallthatthewaveformofanarbitraryorthonormal

modulationcanbeexpressedas

LetX(t)

be

the

first

signal

waveform,

as

Let beanadditionalsetoforthonormal

functionssuchthattheentireset spansthespaceofrealL2

waveforms Successivesignalwaveformscanbeexpandedintermsof

Thereceivedsignalcanbeexpandedas

Y(t)isassumedtobethesumofthesignalX(t),thewhiteGaussian

noiseZ(t)andcontributionsofsignalwaveformsotherthanX

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Successive

transmission

with

arbitrarymodulation


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DetectionwithArbitraryModulation

Denotethecontributionsfromotherusersandsuccessive

signalsfromthegivenuseras

Denote

wehave

Theobservationisasamplevalueof(Y,Y).AssumingX,Z,Z,

andVareindependent,thelikelihoodis

thelikelihoodratiois

ELEC5360 18Y

is

a

sufficient

statistic

for

a

MAP

detector

on

XYisirrelevanttothedecision


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TheoremofIrrelevance

Let beasetoforthonormalfunctions.Let

be

the

input

to

a

channel

and

the

corresponding

noise,

and

Let

andforeachm>n isindependentofthe

pairX and

Z

LetY=X+Z.

ThentheLLRandMAPdetectionofX fromtheobservationof

Y,Y

depends

only

on

Y.

TheobservedsamplevalueofYisirrelevant

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AWGNChannel

Fromlastlecture,wehave (afterdemodulation)

Theresidualnoise isindependentof

Let

thenYisasufficientstatisticsfortheoptimumdetection

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Ourfamiliar

discrete

time

signalmodel!


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MaximumLikelihood(ML)Detector

TheMAPdetector

Typicallythe

prior

probabilities

are

equal,

i.e.,

Hence,optimalrule istomaximizethelikelihoodofr

givenai

or foralli. Thus,wegettheMaximum

Likelihood orML detector

Equivalently,

the

threshold

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DecisionRegions

WithaMDdetector,realNspace ispartitioned

intoMdecisionregions

consistsof

the

received

vectors

that

are

at

least

as

closeto astoanyotherpointin

Thesedecision

regions

are

also

called

as

Voronoi regions

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ModulationwithMemory

Maximum

Likelihood

Sequence

Detector

(MLSD)

Whensignalhasnomemorysymbolbysymboldetectionis

optimum Whenthereismemorytheoptimumdetectormustusethe

observedreceivedsequence

ConsiderasinglebinaryPAMreceivedsignalatkth signal

interval

SincethisisaGaussianrv theconditionalpdf (conditionedon

inputsignalsm

)is

kk nEr

2

2

2

)(exp

2

1)|(

Ersrp kmk


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MLSD

SincethenoisesamplesatdifferenttimeintervalsareindependentthejointPDFofthetransmittedsequencesis

Receiverthatmaximizestheconditionalprobabilityis

MLsequence

detector

BytakinglogsMLsequencedetectorselectsthesequencethatminimizestheEuclideandistance

metric

K

kkkK srpsrrrp

121 )|()|...,(

K

k

kk srsrD1

2),(


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MLSD

Infindingtheminimumitmayappearthatweneed

tosearchthrougheverypossiblesequence

ForNRZI

(Non

Return

to

Zero

Inverted)

which

uses

binary

modulationthetotalnumberofsequencestocheckis2K

WillthereforeneedtocalculateK2Kbranchmetrics

TheViterbi

algorithm

is

asequence

trellis

search

algorithmforperformingMLsequencedetection

thatreducesthesearchrequired

Letsconsider

NRZI

to

see

how

it

works


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MLSD

ConsiderthetrellisdiagramforNRZIinitiallyinstateSo

Memoryis1bit(L=1)trellisreachessteadystateover2

transitions After2Ttwosignalpathsentereachnodeandtwosignal

pathsleave


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ViterbiAlgorithm

At2TfornodeSo thetwometricsforbothsequencesare

TheViterbialgorithmcomparesthese2metricsand

selectsthe

lowest

one and

is

known

as

the

survivor

Eliminationofallotherpathstothatnodeisoksinceanypathbeyond2Tmustusethatsurvivortoremainminimumaswell

Similarlyfor

node

S1 wehave

22

210

22

210

)()()1,1()()()0,0(

ErErD

ErErD

22

211

22

211

)()()0,1(

)()()1,0(

ErErD

ErErD


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ViterbiAlgorithm

Metricsfor3TatnodeSo are(assuming(0,0)and(0,1)aresurvivorsfrom2T)

Metricscomparedagainandsurvivorfound

SimilarlyfornodeS1

Thereforethe

number

of

paths

searched

is

reduced

to

asearchof4ateachstageonly

Thereforetotalnumberofbranchmetricscalculatedis4kratherthank2k

Thisis

avery

good

reduction

in

computational

complexity

2310

2300

)()1,0()1,1,0(

)()0,0()0,0,0(

ErDD

ErDD


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ViterbiAlgorithm

InVAitisunclearhowtoselectasequencetooutput

IfwehaveadvancedtosomestageKwhereK>>Lin

thetrellis

Wefindthatallsurvivingsequenceswithprobability

approachingunitywillbeidenticalinsymbol

positionsK5L

and

less

Inpracticethedecisionsareforcedforallsymbols

afteradelayof5Lsymbolsandhencesurvivingpaths

aretruncated

to

length

5L


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Update

Haveconsidered

Signalspaceconcept

ModulationNoisemodel

Wewill

now

consider

Optimaldetection


Channelcapacity

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TheErrorProbability

Errorevent

fallsoutsidethedecisionregion

or,the

noise

falls

outside

the

translated

region

Theprobabilityofdecisionerrorgiventhat istransmittedis

thereforegivenas

Theaveragesymbolerrorprobabilityis

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PairwiseErrorProbabilities

Thepairwiseerrorprobability isthe

probabilitythat istransmittedwhile isdetected

Itis

given

by

where

sothepairwiseerrorprobabilityonlydependsonthe

distance andthenoisevariance

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Proofof .

Theerroreventis

thereceivedvectorr iscloserto thanto

equivalently,

FromthepropertyofGaussianvectors,wehave

Thenwe

can

get

the

result

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Thecomplexcasecan

beconsideredsimilarly


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UnionBound

Performanceevaluation ofMary modulation:

Errorprobabilitycomputationisquitecomplicated

Engineerstypically

look

for

approximations

which

make

the

system

analysisanddesignlesscomplicated

UnionBound willbedeveloped

Acommon

tool

used

by

communication

engineers

Veryeasytoderive

Cangiveaccurateprobabilityestimates

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UnionBound

Theelementaryunionboundofprobabilitytheory

Unionbound

of

LetD denotethesetofdistancesbetweensignal

pointsin ,and asthenumberofsignalsat

distancedfrom .Thentheunionboundcanbe

writtenas

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PairwiseLowerBound

AstheQfunctiondecreasesexponentiallyas ,

thefactor willbelargestfortheminimum

Euclideandistance

Ifthereisatleastoneneighbor atdistance

from

,then

we

have

the

pairwise

lower

bound

Thislowerboundandtheunionboundhavethe

sameexponent

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Example UnionBound

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Example UnionBound

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Example UnionBound

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AnalysisofOrthogonalModulation

Orthogonalsignalset ,MrealorthogonalM

vectors,eachwithenergyE

Withoutlossofgenerality

Atthereceiverside,thesufficientstatisticsis

TheMLdetectoris

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OrthogonalSignalSets ExactAnalysis

Correctdecisionprobabilityisgivenby

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SymbolErrorstoBitErrors

Symbolerrorsaredifferentfrombiterrors.

Whenasymbolerroroccursallk=logM bitscouldbeinerror

Fororthogonal

modulation,

when

an

error

occurs

anyone

of

the

othersymbolsmayresultequallylikely

Thus,onaverage,halfthebitswillbeincorrect

Thatis,k/2 bitsinerrorforeverykbitswillonaveragebeinerrorwhen

thereisasymbolerror

Hence,foraparticularbit,theprobabilityoferrorishalfthesymbolerror

whenM islarge

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ExactDerivation

Inorthogonalmodulation,whenthereisanerroritwillleadtoanyoneoftheotherM1=2k1 possiblesymbolsequally.Thatis,whenthereisanerror

event,theprobabilityofaparticularsymbolgettingthaterrorisPe/(M1)

For

this

given

symbol

error,

assume

there

are

n bits

in

error

Thereare(k,n)combinationsinwhichthismayhappenandtherefore(k,n)

symbolsintotalwithapossiblen biterrors.

Thus,theprobabilityofan biterrorsoccurringis

Hence,foreverykbitstherewillbeonaverage biterrors

Therefore,

andforlargeM

)1(

M

P

n

ke

)1(1

M

P

n

kn e

n

n

eb PP

2

1

ee

n

n

eb PM

M

M

P

n

kn

kPP

)1(2)1(1

1

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Probabilityofbiterrorforcoherent

detectionoforthogonalsignals.

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OrthogonalSignalSets AUnionBound

Theunionbound

leadsto

kP

N

E

e

b

as0then

dB42.1~39.12ln2If0

Reliable communication!

dB6.1~2lnislimitShannonThe

ioncommunicatreliablefornecessarynotbutsufficientisdB42.1thatseewillLater we

0

0

N

E

N

E

b

b

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Update

Haveconsidered

Signalspaceconcept

ModulationNoisemodel

Wewill

now

consider

Optimaldetection


Channelcapacity

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ChannelCapacity

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ChannelCapacityC:themaximumrateofreliable

communications,i.e.,witharbitrarilysmallerrorprobability

Withoutnoise(andotherdistortions)

Infinitecapacity

Wecantransmitanarbitraryamountofinformationwithareal

number!

Withoutpower

constraint

on

the

transmitted

symbol

Infinitecapacity

Wecantransmitanarbitraryamountofinformationwithlargeenough

transmitpower,supportinglargeenoughconstellation

Tomake

the

problem

meaningful,

we

need

to

consider

practicalconstraints!


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ChannelCapacity

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Withaninputconstraint,withnoise,wewillhavefinite

channelcapacity

CapacityofbandlimitedAWGNChannel(bandwidthW,noise

spectraldensityN0/2)

Nextwe

will

get

to

it

from

CapacityofdiscretetimeAWGNchannel(spherepackingargument)

ContinuoustimeAWGNchannel

h l


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DiscretetimeAWGNChannel

Letthechanneloutputattimekas

noise ,independentofsignalsandofnoiseatallother

times

Inputconstraint

kbitsincomingsourceismappedintoanntuplecodeword

Thecode

rate

is

R

=k/n

Thecapacityofthischannelis

ForanyrateR


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SpherePackingfortheGaussianChannel

Foranycodeword oflengthn:

Thereceivedvectoris

Withhigherprobability,itliesinaspherearoundthesentvector

Thenoisepowerisverylikelytobecloseto

Theradiusofthesphereis (LLN)

Allreceived

vectors

have

energy

no

greater

than

,so

theylieinasphereofradius

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S h P ki f th G i Ch l


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SpherePackingfortheGaussianChannel

Thevolumeofanndimensionalsphereis

Therefore,themaximumnumberofnonintersecting

decodingspheresisnomorethan

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B d li it d AWGN W f Ch l


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BandlimitedAWGNWaveformChannel

Consideratimeinterval[0,T]

Wehave2WTdegreesoffreedom,i.e.,wecantransmit2WTsymbols

TotalenergyisPT,soforthekth symbol

Totalnoise

power

is

,and

each

of

the

2WT

noise

samples

hasvariance

Sothecapacityforeachsampleis

Asthereare2Wsamplespersecond,thecapacityofthe

channelcanbewrittenas

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B d li it d AWGN W f Ch l


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BandlimitedAWGNWaveformChannel

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HighSNR(SNR>>1)

C~Wlog P

Bandwidthlimited,i.e.,increasingPdoesnothelpmuch

Multilevelmodulation

LowSNR(SNR1)

Wehave

Powerlimited

Binarymodulation

Bandwidth(Spectral)Efficiency


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( p ) y

vs.

Power

Efficiency Forreliablecommunication

Define

bandwidth

efficiency

as

r=R/Wand

Wehave

Theminimumvalueof forreliablecommunicationis

obtainedbylettingr 0

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Shannonlimit

Achieving Channel Capacity with Orthogonal Signals


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AchievingChannelCapacitywithOrthogonalSignals

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Withunionbound,wehaveseethattheerrorprobabilitycan

bemadeassmallaspossibleif

However,unionboundisnottight

Thisis

not

the

smallest

lower

bound

on

Usemoretightbounds,wecanprove(Ch6.6Proakis or8.5.3

Gallager)

Mary orthogonalmodulationachievesthecapacityofinfinite

bandwidthAWGNchannel!


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TradeOffs


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Trade Offs

PowerLimitedSystems:Power scarce butbandwidthavailable

ImprovePb byexpandingbandwidth(foragivenEb/N0 )orrequiredEb/N0 can

bereducedbyexpandingbandwidth(foragivenPb)

Multidimensional

signals:

orthogonal,

bi

orthogonal,

simplex

BandwidthLimitedSystems:bandwidthscarce

Whenwetransmitlog2(M) bitsinTsecusingabandwidthofWHz,then

Bandwidthefficiency: R/W=log2(M)/WTbits/sec/Hz

ThesmallertheWTproductis,themorebandwidthefficientwillthe

systembe

MaximizeR overthebandlimited channelattheexpenseofEb/N0 (fora

givenPb)

Bandlimitedsignaling:ASK,PSK,QAM

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Summary


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Summary

Optimaldetection

Binarydetection

Arbitrarymodulation


Channelcapacity

Readingassignment

Ch8ofGallager

Ch4,6.5

6.6,

of

Proakis

Checkoptimaldetectionanderrorprobabilityofdifferent

modulationschemes

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Documents

Lecture 4-Detection, Performance Analysis