lecture 4-detection, performance analysis
TRANSCRIPT
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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
Errorprobabilityanalysis
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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AdditiveGaussianNoise
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AdditiveGaussianNoise
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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
Errorprobabilityanalysis
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
Errorprobabilityanalysis
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
Errorprobabilityanalysis
Channelcapacity
Readingassignment
Ch8ofGallager
Ch4,6.5
6.6,
of
Proakis
Checkoptimaldetectionanderrorprobabilityofdifferent
modulationschemes
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