Nouvelles bornes et techniques de synchronisation

Universitddu Qu6bec
lnstitut national de la recherchescientifique
EnersieMat6riauxT6l6communications
Nouvellesborneset techniquesde synchronisation
temporellepour les futurs systdmesde
communications
sansfil
Par
AhmedMasmoudi
M6moirepour I'obtentiondu gradede Maitre bs sciences(M.Sc.)
en tdl6communications
Jury d'6valuation
Directeurde recherche
SofidneAffes,INRS-EMT
Examinateurinterne
SergeTatu,INRS-PUT
Examinateurexterne
Paul Fortier,
D6partementde g6niedlectriqueet de g6nieinformatique
Universit6Laval
@ droitsrdserv6sde AhmedMasmoudi,2012.
{ijl
a>'Jl ,'!U?'Jl
.-rt
.
J
v't
:
I
Avant-Propos
Je souhaite remercier Pr SofieneAffespor,trm'avoir accept,6dans son /quipe et guid6 tout
au long de ma maitrise. Sesremarques et critiques pertinentes m'ont conduit vers Ia bonne
voie. Un grand merci aussi d M. Faouzi Bellili. C'6tait un vrai plaisir de travailler avec lui.
Je remercie aussi mes amis Mouhamed, Hadhami, Foued, Ghaya et mes camarades de
I'INRS pour leurs encouragementset leurs soutienstout au long de ces deux anndesainsi
que Jigo et Bosspour leur APS.
Ces avant-propos seraient incomplets sans un remerciement adressd aux membres de ma
famille, en particulier mes parents, monfrdre Mouhamed et ma saur Ines. Ce travail leur
appartient d tous.
.6ri| diq 5j
R6sum6
Ce m6moire est consacr6d la synchronisationtemporelle(ou estimationdu d6lai) pour
les r6cepteursnum6riques.Les principalescontributionsincluent deux grandsvolets.Le
premier consisted estimerle retardoi les nouvellestechniquesd'estimationd6velopp6es
sontbas6essur le critdredu maximumde vraisemblance.D'une part, nousd6velopponsun
estimateuri maximum de vraisemblancepar la m6thode"importancesampling"pour les
signauxnum6riqueslin6airementmodul6soir I'on considdreun seultrajet de propagation,
et doncun seulparambtreh estimer.Danscetteconfiguration,noussupposonsque les donn6estransmisessont totalementinconnuesau niveaudu rdcepteur.Le d6lai resteconstant
sur I'intervalled'observationet le bruit est suppos6blanc.Nous appliquonsaussila m6thodepropos6edansle casd'un seultrajet au cas de plusieurstrajetset donc le nombre
de paramdtresi estimeraugmenteavecle nombrede trajetsd6tectds.Nous signalonsque
dansce cas le signal transmisest connupar le r6cepteur.Cette mdthodepeut s'appliquer
dansles radarset les systBmes
En plus nousnoussommesint6ress6s
de localisations.
d la
synchronisationtemporellepour les systbmesCDMA en ddveloppantdeux estimateursd
maximum de vraisemblance.Le premier sebasesur la m6thode"importancesampling"et
I' autresur l' algorithmeit6ratif "expectationmaximization".
D'autre part, nous d6rivonsaussiles expressionsanalytiquesdes bornesde Cram6r-Rao
pour les estimateursnon biais6sdu retard dans le cas des signaux QAM carr6set des
systdmes
CDMA.
Tabledesmatibres
Avant-Propos
R6sum6
ll
List des figures
vt
Introduction
I
La synchronisation temporelle pour les signaux modul6s
3
1.1 Introduction
4
I.2
Uimportancede la synchronisation
temporelle
4
1.3 Estimateuri maximumde vraisemblance
1.4 Estimateursdu retardi maximum de vraisemblance.
pour FaibleRSB
1.4.I Estimateur
6
7
7
9
I.4.2
Estimateurd Maximum de VraisemblanceConditionnel
1 . 5 Limites de performancedesestimateurs
9
1.5.1 LesBornesde Cram6r-Rao.
9
l0
1.5.2 Les Bornesde Cram6r-Rao
Modifi6es
I2
1 . 6 Conclusion
A Non-Data-Aided Maximum Likelihood Time Delay Estimator using Importance Sampling
15
2.I
16
Introduction
2.2 Discrete-TimeSignalModel
18
2.3 Likelihoodfunction
19
2.4 GlobalMaximizationof the Compressed
LikelihoodFunction
2I
2.5 The ImportanceSamplingTechnique.
22
2.6 Estimationof the Time Delay
2.6.1 First scenario: r* € 10,PTI
25
2.6.2 Secondscenarioi r" e [0,7] .
2.6.3 Summaryof steps
25
28
29
.30
2.8 Conclusion
34
Closed-Form Expressionsfor the Exact Cram6r-Rao Bounds of Timing Recovery Estimators from BPSK, MSK and Square-QAM Transmissions
41
3.1 Introduction
43
3.2 SystemModel
44
3.3 Time DelayCRLB for SquareQAM-ModulatedSignals
46
3.4 CRLB for BPSK andMSK ModulatedSignals
52
3.5 GraphicalRepresentations
53
3.6 Conclusion
56
A Maximum Likelihood Time Delay Estimator in a Multipath Environment
Using Importance Sampling
65
4.I
66
Introduction
4.2 SystemModel andCompressed
LikelihoodFunction
68
4.3
69
Global Maximization of the CompressedLikelihood Function
4.4 The ImportanceSamplingBasedTime DelaysEstimation .
7T
4.4.1 IS Concept
7T
4.4.2 Time DelaysEstimationin Active Systems
75
'77
4.4.3 Time DelaysEstimationin PassiveSystems
4.5 SimulationResults
79
4.6 Conclusion
82
Maximum Likelihood Time Delay Estimation for Direct-SequenceCDMA Multipath Ttansmissions
87
5.1 Introduction
88
5.2 SystemModel andBackground.
9I
5.3 The Crambr-Rao
Lower Bound
93
95
98
103
105
108
5.4 ExpectationMaximizationAlgorithm
5.5 The ImportanceSamplingTechnique.
5.6 ExtensionTo MC-CDMA Systems
5.7 SimulationResults
5.8 Conclusion
Conclusion
118
Table desfigures
l.l
Diagrammede l'mil pour un filtre d cosinussur6lev6avecun coefficient
de retomb6ede 20%
t . 2 Diagrammede I'eil pour un liltre d cosinussur6lev6avecun coefficient
de retomb6ede 100% .
2.I
Plot of eto,( ) for p\ : 20 andp\:
f o rK :
10 usinga root-raisedcosinepulseand
24
100.
2.2 Plot of the cumulativedistributionfunction (CDF), G'(r) whosepdf is
g'(r),SNR=5dB.
2.3 Performance
versusp', for SNR=10dB.
26
30
2.4 Estimationperformance
consideringL'.,00?)I g'(r) andsettingL|,ooQ)I O'?)
31
to 1 with QPSK modulation
2.5 Comparisonbetweenthe estimationperformanceof the IS algorithmusing
the two scenariosand the tracking performanceof the CML-TED using
QPSKmodulation
2.6 Comparisonbetweenthe estimationperformanceof IS algorithm and the
tracking performanceof CML TED using QPSK modulation and for a
32
11
roll-offfactor of 0.2.
JJ
2.7 NormalizedMSE of the time delayestimatefor differentQAM modulation
order,usinga root-raisedcosinefilter with a roll-off factor0.5. .
34
2.8 Comparisonof the estimationperformancewith and without forced symbols using 16-QAM modulationandfor a roll-off factorof 0.5.
2.9
P l o to f s o ( . ) .g t ( . ) a n d9 2 ( . ) .
34
36
3 . 1 Compressionbetweenthe empirical CRLB and the analyticalexpression
in (3.44) for different modulation orders using K :
cosinepulsewith rolloff factorof 0.2.
100 and a raised-
53
3 . 2 CRLB/IVICRLBratio vs. SNR for differentmodulationordersusingK :
pulsewith rolloff factorof 0.2 and 1.
100and a raised-cosine
3 . 3 CRLB vs. SNR for different modulation orders usins K : 100 and a
pulsewith rolloff factorof 0.2.
raised-cosine
54
55
3 .4 CRLBA4CRLB ratio vs. SNR for differentmodulationsanda time-limited
shapingpulse.
55
4.I
Complementary
cumulativedistributionfunctionof the ratioK(r*,n).
4.2 Plotof gr, (.) at SNR: 10for (a)p\:1 and(b) p\:6.
t5
4.3 Estimationperformanceas a function of p',
4.4 Estimation performanceof the IS-based,EM ML and the MUSIC-type
algorithmsin an activesystemvs. SNR.
4 .5 Estimation performanceof the IS-based,EM ML and the MUSIC+ype
algorithmsin an passivesystemvs. SNR.
4 . 6 Estimation performanceof the IS-based,EM ML and the MUSIC-type
algorithmsin an passivesystemasa functionof Ar.
75
80
81
82
83
4.7 Estimation performanceof the IS-based,EM ML and the MUSIC-type
algorithmsvs. signalbandwidthat SNR: 10 dB in an activesystem.
4.8 Estimationperformanceof the IS-basedand the MUSIC-type algorithms
83
84
vs. Dopplershift at SNR: 10 dB.
5 . 1 Estimationperformanceof theIS-based,theEM-basedandthe Root-MUSIC
algorithmsfor closely-separated
delays,M : 4.
106
5 .2 Estimationperformanceof theIS-based,theEM-basedandthe Root-MUSIC
for closely-separated
delays,AI :7.
5 .3 MSE vs. number of antennabranchesM for ly' :
107
1 symbol. K :
subcarrier,andSNR: 10 dB.
5.4 MSE vs. numberof receivedsymbolsl/ for M : I antennabranch,K :
1 subcarrier.
and SNR: 10 dB.
L
108
108
5 .5 MSE vs. numberof subcarrierwith M : 7 and N : 1.
5 .6 Complementarycumulativedistributionfunction of the ratio F (A,r^,,).
vi
109
111
Introduction
La synchronisation
est une tacheessentiellepour n'importe quel systdmede communicationsnum6rique.Souventles performances
d'un systbmede transmissionsont dict6es
par la fiabilit6 de la fonction de synchronisation.En effet, le signalreguest compldtement
connu d l'exception desdonndestransmiseset desparamdtresintroduitspar le canal (d6lai, phase,ddcalagefr6quentiel...).M6me si la fonction primordiale du r6cepteurest de
reconstruireles donn6estransmises,ceci ne peut sefaire qu'en connaissantles paramdtres
introduitspar le canal.Dansce travail, nousnousint6ressonsi la synchronisationtemporelle,ou en d'autrestermesd I'estimationdu d6laide propagation.Dansce contexte,bien
que plusieursestimateursdu d6lai aient €t6 d6velopp6sdurant les dernibresd6cennies,
ce problbmesusciteencoreune grandeattentionsurtoutaprbsle succbsdes systbmesde
communicationsansfil et les avanc6esen micro6lectroniquequi offrent plus de possibilit6
d'impl6mentation.
semiLes estimateurs
du retardpeuventOtreclass6sen plusieurssous-classes
: supervis6s,
aveuglesou aveugles.
Les techniquessupervisdes
dessymboles
exploitentla connaissance
transmisdans les blocs de synchronisationpour faciliter la proc6dured'estimation.Les
puisqueles symbolesinm6thodessemi-aveugles
s'inspirentdestechniquessupervis6es
connussontd'abordestimdspuisutilis6sdansla synchronisation.
Bien qu'ellesrequiBrent
la transmissionde moinsde symbolesconnus,ellessouffrentd'erreursde d6tectionsqui
d6gradent
lesperformances
au casoi les
du systdme.Dansla suite,nousnousint6ressons
donn6estransmises
sontinconnues.
Dans ce contexte,plusieursestimateursont 6t6 rapport6sdans la litt6raturepour offrir les meilleuresperformancespossibles.Il est connu que I'estimateurd maximum de
vraisemblance
est un estimateurasymptotiquement
efficaceet qu'il r6alisela meilleure
performancei des valeursrelativement6lev6esdu rapport signal sur bruit (RSB), m6me
sur de courtsintervallesd'observation.Par cons6quent,
il a €t6 I'objet de recherches
intensives.Dansle casoir les symbolestransmissontconnus,uneexpressionanalytiquedu
maximum global de la fonction de vraisemblancepeut Otreobtenue.Toutefois,lorsque
les donn6estransmisessonttotalementinconnues(c'est-d-direle parambtred'int6r0tdoit
6tre estim6 d'une manibre aveugle),la fonction de vraisemblancedevient une fonction
non-lin6aireet il est difficile, mOmeimpossible,de trouverune expressionanalytiquedu
maximumglobal de Ia fonction de vraisemblance.
Dansce cas,desm6thodesde rdsolution
num6riquesdoiventOtreenvisag6es.
Le travail pr6sent6dansce m6moires'inscrit dansle
cadrede d6veloppementde m6thodespour trouver le maximum global de la fonction de
vraisemblance.L 6tapeprimordialepour le d6veloppement
de tels estimateursest d'exprimer le problbmesousla forme d'un modblelin6aireg6n6ralis6.Pource faire noustraitons
les modblesles plus r6pandusen communicationnum6rique,h savoirle casd'une transmissiond'un signalmodul6sur un seultrajet,une transmissionsur plusieurstrajetset le
puisqueles estimateurs
casdes systbmesCDMA. Une telle distinctionest indispensable
par la suitechangentd'un moddlei un autre.
d6velopp6s
Une fois que les estimateurssontd6velopp6s,nous6valuonsleursperformancesen termes
de variancede I'estim6 comme une mesurede performancesdu systbme.Pour ce faire,
il est dvident qu'il faut les compareraux autresestimateursmais aussipar rapport i une
borneinf6rieurede la variancede tout estimateur.Dansce contexte.les bornesde Cram6rRao sont connuescomme des bornesinf6rieurescontre lesquellesles performancesdes
estimateurssont compardes.Elles indiquent la limite inf6rieurede la pr6cisiond'estimation qui peut Otreatteinte.Notre revuede la litt6raturenous a r6v6l€,que cesbornesn'ont
pas encore6t€ d€rivdesanalytiquementen estimationaveugle.Elles n'ont 6t6 calcul6es
que de fagon num6riqued partir d'expressionstrbs complexes.La ddrivationde ces expressionspermetde mieux caract6riseret analyserles performancesdu systdme.Dans ce
travail,nousd6rivonsles expressionsanalytiquesde cesbornespour les estimateursde d6lai en pr6sencede la pluspartdesconstellations
utilisdescourammentet pourles systbmes
CDMA.
Ce rapport est structurdcommesuit. Dans le chapitre1, nouspr6sentonsune brbveintroductionsurl'estimationdu retarden communicationsansfiI. Les contributionseffectu6es
sont pr6sent6esdansles chapitressuivant.Le chapitre2 d1crit le nouvel estimateurpour
les signauxmodul6s.Ensuite,dansle chapitre3, nousd6rivonsles expressions
exactesdes
bornesde Cramdr-Raopour l'estimationaveugle.Dansle chapitre4, nousd6veloppons
un estimateurde d6lai dansle cas d'un canali plusieurstrajets.Et dansle dernierchapitre nousnousint6ressons
aux systbmes
CDMA oi nousd6veloppons
deuxestimateurs
d
maximum de vraisemblanceet les bornesde Cram6r-Raocorrespondantes.
Chapitre 1
La synchronisationtemporellepour les
signauxmodul6s
1.1 Introduction
La synchronisation
est une tdcheessentiellepour n'importe quel systbmede communicationsnum6rique.La synchronisationtemporelle,appeldeaussiestimationdu d6lai de
propagation,est un problbmefr6quemmentrencontr6dansla synchronisation.Le but est
de s'assurerque les 6chantillonspris du signal regu concordentavecla valeur optimale
pour reconstruireles donn6esd'une manibrefiable. Autrementdit, le ddlai introduit par le
canal de propagationdoit Otrepris en consid6rationau niveaudu r6cepteur.Une des solutions pour estimerce paramdtreest d'envoyerun signal connu au r6cepteur.La mdthode
classiqueconsisted faireI'auto-corrdlation
de I'observation.
Cependant,
cetteapprocheest
puisquele signaltransmisne transporte
co0teuseen termesd'6nergieet debandepassante
pasd'informationutile.C'estpourcetteraisonqueplusieurstravauxont traitdle probldme
de la synchronisationtemporelleen utilisant directementle signalregu.Les m6thodes16sultantessontclass6es
oir le r6cepteurn'a pasbesoinde
commeaveugle(non-data-aided)
connaitreles donn6stransmises.
Une 6tapeimportantedansle processusd'estimationest
de d6terminerune fonction objective,calcul6ed partir du signalregu,de telle fagonqu'une
estimdedu d6lai de propagationpeut Otreobtenu.En se basantsur cette fonction, les estimateursdu d6lai sont class6scomme suit [1] : d erreur quadratiquemoyenneminimale
(minimum mean squareerror), forgaged z6ro (zero forcing), early-lategate, maximum
de vraisemblance,
au critdredu maximumde
etc.Dansce qui suit,nousnousint6ressons
vraisemblance.
1.2 Ltimportance de la synchronisationtemporelle
Dansplusieurssystbmesde communications,les performancessont6troitementli6es i la
margetemporelleallou6e.La margetemporelleest I'erreur de synchronisationmaximale
que le rdcepteurtolbre sansprovoquerdes d6gradationsen performances.Elle peut 6tre
jug6e en examinantle diagrammede l'eil du signali l'entr6edu bloc de ddcisionau
plusieursr6pliquesdu signal
r6cepteur.
Le diagrammede I'eil estobtenuen superposant
reEu.Le nom de ce diagrammevient du fait que la figure obtenueressembled un ceil
humain.Pour 6valuerla margede mancuvre d'un systdmepar rapportaux distorsionsque
subit le systdme,par exempleb I'erreur de synchronisation,h l'interfdrenceinter-symbole
et au bruit, nous6tudionsla forme et I'ouverturedu diagrammede I'eil.
Les figuresl.l et I.2 illustrentles diagrammes
de I'ail en absencede bruit et en utilisantun filtre d cosinussur6lev6avecun coefficientde retomb6de 20%et I00To,respectivement.Uinstantauquell'ouverturedu diagrammeest maximalecorrespondd l'instant
Commenouspouvonsle voir, I'ouverturede l'ail diminuede
optimald'6chantillonnage.
plus en plus que nous nous 6loignonsde I'instant t : 0. En effet, f interf6renceinters'6loignede I'instantopsymboleaugmented'autantplus queI'instantd'6chantillonnage
Donn6es
-0.6
binairc, un c@ffcient
-0.4
-o.2
0
de relomb6e
0.2
de 20%
0.4
D6calage tempore
FrcuRe 1.1- Diagrammede I'eil pour un filtre d cosinussur6lev6avecun coefficientde
retomb6e
de20%
Donn6es binaire, un c@t{icient
-0.8
-0.6
-0.4
-o.2
de retomb6e
0
0.2
D6calage temporel
d€ 100%
0.4
FIcune 7.2 -Diagrammede I'eil pour un filtre d cosinussur6lev6avecun coefficientde
retomb6ede 100%
la margede bruit diminue.En absencede bruit, les donn6es
timal, et par cons6quence,
peuventOtreparfaitementd6tect6estant que I'instant d'6chantillonnageest d I'intdrieur de
la zoned'ouverturede I'eil. Une synchronisationparfaite qui satisfaitle critdre de Nyquist minimise ou annulecompldtementI'interf6renceinter-symbole.Cependant,plus le
rapportsignalsurbruit (RSB)diminue,plus l'intervalled'6chantillonnage
fiablediminue
et le systbmedevientplus sensibleaux effeursde synchronisation.
1.3 Bstimateur e maximum de vraisemblance
Uestimateurir maximumde vraisemblance(MLE) estun estimateurdit h efficacitdasymptotique.Il est d6fini commela valeur du paramdtrequi maximisela fonction de vraisemblance.En g6n6ral,rl a €t6 prouv6 en l2l que le MLE est asymptotiquementnon-biais6,
atteintla bornede Cram6r-Rao(CRLB) et sonerreurpossddeuneDistributionGaussienne.
Concernantnotreproblbme,leMLE devraitatteindrela CRLB pour de grandesvaleursdu
rapport signal sur bruit. Pour cette raison,la plupart des algorithmesde synchronisation
sontbas6ssur le critbrede maximum de vraisemblance.
La formulation du problbmed'estimationvarie suivantque nous consid6ronsun signal h
tempscontinu ou un signal d tempsdiscret.La premibreapprochesembleOtrela plus appropridei causede la naturephysiquedu signal,mais les r6cepteursnumdriquesopdrent
sur dessdquences
6chantillonndes.
Danscettepartie,nousconsid6ronsd'aborduneformulationi tempscontinuepour 6tendre,
dansles chapitresqui suivent,I'approcheau tempsdiscret.Nousnotonspar 7 l'ensemble
des parambtresinconnusqui inclut la fr6quenceporteuse,l'offset de phase,le retard introduit par le canalet les symbolestransmisdansle cas d'une estimationaveugle.Nous
adoptonsla notationr(t,-f) pour le signalreguen absencede bruit qui met en 6videncela
d6pendance
en 1 . Le moddleen bandede baseest :
y(t):r(t,'Y)+w(t),
(1.1)
on ru(t) est le bruit additif complexe.On considbreque g(f ) estune r6alisationd'un processusal6atoirey(l) pour une valeurdonn6ede I : 7. En effet,une r6alisationde y(f )
a un certaindegr6de ressemblance
avecg(/) d6pendamment
de la ressemblance
entre
et t(t,.y), en d'autrestermes,la distanceentreI et 7. Uestimateurd maximum
"(t,7)
entrey(t) etla
de vraisemblance
estbas6sur le calcul de ] de sorteque la ressemblance
r6alisationy(t) soit maximale.En termesde probabilit6,nousappelonsp(V(t)17)la dene pr 7 .Supposonsquepour deuxr6alisationsde ],
sit6de probabilit6de y(l) conditionnd
not6espar7t et ]2, nousavons:
p\(t) : a(t)17')< p1(t) : a(t)172),
(t.2)
que11.
alors12 estdit plus vraisemblable
CommenousI'avonsmentionn6,le but estde maximiserp(y(l) : a(t)17) parrapportd l.
La position du maximum est appel6eestim6d maximum de vraisemblanceet est donnde
par:
i*":
: a(t)17)}
argmax{p(y(l)
-Y
(1.3)
Cependant,I inclut, en plus des parambtresde synchronisation,les symbolesinconnus
qu'on ne cherchepas d estimerd ce niveau.C'est pour cetteraisonque les paramdtres
d
estimersont rassembl6sdans,\ et les autresparambtres,appel6sparambtresde nuisance,
sontrassembl6s
dansn. Maintenant,nous devonsreformulerI'estimateuren (1.3) pour
tenir comptede n. Soient\ et fi, deux valeurshypoth6tiquesde ), et n, respectivement.
En mod6lisantn commeun vecteuraldatoirede densit6de probabilrt€p(n),la formule
desprobabilit6stotalespermetd'6crire :
floo
p(v(t): a(t)l,\): /
Jx
p@)an.
p(v(t): a(t)17)
(1.4)
Puis,I'estimateurd maximumde vraisemblance
de A est :
\u r:
a r gm ax{ p( y(:r ) g( r ) l^) }
(1 . s )
I
Dansla suitede ce chapitre,ainsique dansles chapitres2 et3, nousnousconcentrons
sur
le casoir ) estun scalaire(le d6lai r) et noustraitonsle casoU .\ estun vecteurdansles
chapitres4 et 5.
1.4 Bstimateurs du retard i maximum de vraisemblance
1.4.L Estimateurpour FaibleRSB
Dans cette section,nous nous int6ressons
au cas aveugle(symbolesnon connus).Nous
pr6sentonsle traditionnel estimateurpour faible valeurs de RSB d6velopp6en [3]. La
fonction de vraisemblanceest :
: exp
y(t)r(t)dt- ,'AVr},
/t(al,,c)
*al,'
{fr l,'
(1.6)
oi 1/0 est la puissancedu bruit, c : {co, cr, ..., c;} repr6sente
de donn6es
la s6quence
inconnueet r(t\ estd6finiecommesuit :
(r.7)
i:o
L objectifestd'estimerle retardT sansavoirune connaissance
a priori dessymboles.La
m6thodela plus directe est de consid6rerles donn6esc comme un vecteuraldatoire.La
fonction de vraisemblanceest moyenn6epar rapportaux donn6estransmisespour obtenir
la fonction de vraisemblanceinconditionnelle:
A(slt): E"{A(rlr,
c)}.
(1.8)
Malheureusement,l'expressionanalytiquede la fonction de vraisemblanceinconditionnelleestdifficile bi6valuer.C'estpour cetteraisonquequelquesapproximations
sontutilis6es.Premidrement,le secondtermedansI'expressionde la fonction de vraisemblanceen
(1.6) est ignor6.Cettesimplificationentraineraune ddgradationdesperformances
d'estimation. Ensuite,une approximationde la fonction rdsultantepeut Otreenvisag6een utilisantle d6veloppement
de Taylor sousI'hypothbsede faiblesvaleursde RSB. Alors nous
obtenons[3] :
x L+ ,{r),{r)dt
ty(al,,c)
+hll,' u@,(Dotf
fr lo'
(1.e)
Considdrantla d6finition(1.7),nousavons:
- ir - r)d't
lo',t),tDo,:E,n
lo'v(t)h'(t
(1.10)
Troisibmeapproximation,puisqueles coefflcientsde h(t) sont n6gligeablespour t f
[0,7], les limitesde l'int6graledans(1.10)peuvent6tre6largiesd I'infini pour obtenir:
L-1
7T
I a U ) r ( t ) dxt l c ; r ( i T * r ) '
(1.11)
JoE
r ( 1 ):
/^+-
I
a(r)h(u-t)du.
(r.r2)
J-a
(1.11)dans(1.9)et tenantcomptede I'hypothdseE{c6} : g,
Maintenant,noussubstituons
la fonctionde vraisemblance
inconditionnelles'dcritcommesuit :
L_L
L @ l r ) : D r 2 ( i T+ r ) ,
(1.13)
i,:o
oir les termesconstantssont ignor6s.De cette fagon,une expressionplus pratique de la
quadratique
fonctionde vraisemblance
inconditionnelleesttrouv6e.C'est une expression
qui impliquela r6ponsedu filtre adaptlau signalreEu.A partirde (1.13),la valeuroptimale
du retard r maximisel'6nergiede la s6quence{r(i,T + r)}. Pour trouvercette valeur
optimale,il faut annulerla d6riv6ede A(ylr) par rapporti z :
L-7
\y'(al")= 2Dr(i,T + r)r' (tT + r).
(1.14)
i:o
Pourz - rp,la d6riv6edela fonctionde vraisemblance
inconditionnelle
estconsid6r6e
commeune erreurd'estimationet utilis6epour estimerle retardde fagonit6rative :
Tk+L: rx -l p'e(k),
(1.15)
e(k) : r(kr + 16)r'(kr + rk),
(1.16)
avec
et p correspond
aupasd'adaptation.
1.4.2 Estimateur i Maximum de VraisemblanceConditionnel
Nous avonsvu que suite i plusieursapproximations,le critdre du maximum de vraisemblance se formule d'une manidrepratique.Cependant,I'algorithme est ddriv6 sousI'hypothbsed'un faibleRSB.Cettehypothbselimite les performances
de I'estimateurpour les
largesvaleursdu RSB de telle fagonque la diff6renceentrela bornede Cramer-Raoet les
performances
d'estimationest d'autantplus importanteque le RSB augmente.En effet,
I'estimateurI faible RSB est une approximationde I'estimateuri maximum de vraisemblance.A grandRSB, cetteapproximationne refldteplus la vraie fonction de vraisemblanceet les perfonnances
de I'estimateurne sontplus optimales.C'estpour cetteraison
qu'un autrealgorithme,connusousle nom de maximum de vraisemblanceconditionnelle
(MVC), a 6t6ddveloppeg\Afin d'6viterles approximations,
une approchediff6renteest
utilis6epour calculerla fonction de vraisemblance.Contrairementi la premibremdthode
pr6sent6e
en 1.4.1,les symbolesregussontmod6lis6scommed6terministes
et inconnus.
Les parambtresde nuisance(donn6es,phase)sont exprimdsen fonction du parambtred
estimeret du signal regu. De cette fagon,nous obtenonsune fonction qui ddpendseulement du parambtreinconnuqui est le retardh estimer.A partir de cettefonction, un signal
d'erreurestcalcul6puis utilisdpour mettrei jour I'estim6du retard.
Nous pr6sentonsplus de d6tailssur ce point dans chapitre2.
1.5 Limites de performancedes estimateurs
1.5.1 Les Bornes de Cramdr-Rao
Considdrons
n'importequellem6thoded'estimationde r et notonspar ? I'estim6correspondant.Vu que ? d6pendde I'observation'y,diff6rentesobservationsengendrentditr6rentsestim6s.Dansce casi estune variableal6atoiredont la moyennepeutcoincideravec
les vraiesvaleursde r. Dansce cas,l'estimateurest dit non biais6.Cettepropri6t6estun
caractbred'6valuationdesperformances
d'estimationpuisque,en moyenne,l'estimateur
fournit la vraie valeurdu parambtre.Cependant,
l'erreur d'estimationT- r est aussiune
mesureimportante,d'oir la n6cessit6ede minimiser cette erreur,ou encoresa variance.
Alors quandestce qu'on peutdire quel'erreurd'estimationestacceptable
?
Dans ce contexte,la borne de Cram6r-Raoest une limite thdoriquequi fournit une borne
inferieurepour la variancede tout estimateurnon-biais6[2] :
var{?-r}>CRLB(r),
( 1 r.7 )
avec
CRLB(r) :
1
"{{*-m;'1
(1.18)
Dansles probldmesde synchronisation,
I'applicationde cetteborneest difficile i cause
de la complexit6de A(rlr). En effet, nous devonsmoyenner.tt(rlr,u) par rapportaux
parambtresde nuisancet^c:
A(rlr):
I_:
Le/,u)p(u)du
(1.1e)
qui,jusqu'd prdsent,n'a pasd'expressionanalytique.Nouspr6sentons
dansle Chapitre3
une mdthodepour d6riverl'expressionanalytiquede la borne de Cram6r-Raopour l'estimation du d6lai et ceci pour une large gammede modulations.
Une alternatived la borne de Cram6r-Rao,connuesousle nom de borne de Cramdr-Rao
modifide,estpr6sent6e
en [5] pour contournerles probldmesde calcul.
1.5.2 Les Bornes de Cram6r-Rao Modifi6es
Une approchemodifi6e,propos6epar D'Andrea,Mengali et Reggiannini[5] est souvent
utilis6e.I1 s'agit de la bornede CramerRao modifi6e(MCRB pour Modified CramerRao
Bound).Cettenouvellebornes'6crit :
MCRLB :
l/o
E-U{'lltu#ell'"}
(r.20)
Dans(1.20),la moyenneE,{.} esteffectu6esur les parambtres
de nuisanceu et K estle
nombrede symbolesdansI'intervalled'observation.La notationr(t.r. z) est introduite
afin de s6parerle parambtred estimerr desparambtresde nuisancez.
Deux remarquesdoiventOtresoulign6esh ce niveausur la MCRLB. Premibrement,dans
les 6tapesde d6rivationde la borne,nous supposonsque :
- les densit6sde probabilit6de la phaseet de la fr6quencesont connues;
- les symboles{c;} sont des variablesal6atoiresind6pendantes,
de moyennenulle et
E{(c6)2}: g.
De plus, D'Andrea a montr6que la borne de Cram6r-Raoest sup6rieurei la borne modifl6e [5]. Il est 6videntqu'il y a lgalitd si le vecteurt^cest parfaitementconnu.Notez que
cetteremarquerestevalide quelquesoit le parambtred estimer.
La d6rivationde la MCRLB peut ere effectu6ede la fagonsuivante:
10
avec
:
at,
n.
Ion' {y1^'1t7ll'}
".U"'\ry#ell'")
mt '/ (, \t ) : SL- c i p U/ , - i T - r ) ,
(r.2r)
(r.22)
i
dans(1.21)
et p(t) : dh(t)ldt Commem'(t) estind6pendante
de f . et 0,le moyennage
selimite au moyennagesurles symboles6mis.Parsuite,nousobtenons:
p(t- iT - r)
: CI
E-{llm'(t)ll2}
i
(( :\
; \- P'
7t "'\T)
n
:
"i2rn(t-r)/'r
(r.23)
la transformdede Fourierde p2(t). La dernibrelgalitd dans(1.23)
ot &(/) repr6sente
est v6rifi6epar la formulede Poisson.Ensuite,consid6rant(I.2I) et (l.23) et notantque
ffr siz""{t-,)lr 5(n)KT,nous obtenons:
p,(:\
--l/o[*' llar(lrr.u)ll'0,]
u-{
-J : 9r
r/-'"\r/Jo [o',,,nn,,-,)/rd.L
ll o, ll
:
KCP2(0)
(r.24)
Cependant,
de Fourrierde h(t) :
&(0) esten relationdirecteavecH (f),la transform6e
'*
a&!D) ' :
(
p2e)
:
n,.-[ f lH(f )l,df.
S+* d t
\",
J-* \
J-co
/
(r.2s)
Finalement,en int6grant(l .24)dans( 1.20),nousobtenonsle rdsultatd6sir6:
:
MCRLB(r)
#*troE",
(r.26)
oi E" est 1'6nergiemoyennedu signal6misepar symboleet C7 est le carr6de la largeur
de bandemoyennenormalis6equi d6pendde la fonction de mise en forme et sontdonnds,
par :
respectivement,
u" :
r
r+*
(f
,
; J_* lH )l"t.f
\'h:r_,fi f,lH(f)l,df
I=lw)luf
(r.27)
(1.28)
proportionnelle
L'expressionanalytiqueen (1.26)montrequela MCRLB estinversement
au rapportsignalsur bruit E"ll'{o et d la taille de l'observation.De plus,elle est inversementproportionnelle
d,C6,le carr6de la largeurde bandede H(f). Ceci veutdire que
l'estimation du retardestplus facile d effectueravecdessignauxd largebande.Ceci peut
s'expliquerintuitivementpar le fait que les filtres de miseen forme d largebandeont relativementunepetite dur6etemporelleet doncpeuvent6tremieux d6tect6sen pr6sencede
bruit.
L6
Conclusion
Dans ce chapitre,nous avonspr6sent6les caract6ristiques
de l'estimateurd maximum de
vraisemblance.
L'absenced'une expressionanalytiquede I'estimda favoris6le d6veloppementde plusieursalgorithmesd'estimationavecdes complexit6set des performances
vari6es.Dans ce contexte,la d6rivationde bornes de pr6cision d'estimation est importantepuisquecelles-cirepr6sententune limite th6oriquei la performancedesestimateurs.
Les bornes de Cram6r-Raoindiquent les limites inferieurs pour la variancede I'erreur
d'estimation.Cependant,leur applicationau problbmede synchronisationr6sulte en de
s6rieusesdifficult6s math6matiques.
Les bornesde Cramdr-Raomodifi6essont beaucoup
plus faciles d calculer,mais elles se ddtachentdes vraiesbornesde Cram6r-Raode fagon
possibles.
impr6visibleet doncne refldtentpasles vraiesperformances
T2
Bibliographie
t l l J. W. M. Bergmans,Digital BasebandTransmissionand Recording,Boston : Kluwer Academic Publishers,1996.
t21S. Kay, Fundamentalsof StatisticalSignal Processing,Estimationtheory,PrenticeHall, EnglewoodCliffs, NJ, 1993.
Techniquesfor Digital Receivers",New
t3l U. Mengali and A. N. D'Andrea,"Sychronization
York : Plenum,1997.
I4l J. Riba, J. Sala, and G. Yazquez,"Conditional Maximum Likelihood Timing Recovery :
EstimatorsandBounds,"inIEEE Trans.on Sig.Process,vol.49,pp. 835-850,Apr. 2001.
t5l A. N. D'Andrea,U. Mengali, and R. Reggiannini,"The modifiedCramer-Raobound and
its applicationsto synchronizationproblems,"IEEE Trans.Comm.,vol.24, pp. 1391-1399,
Feb.-Apr.1994.
13
Chapitre2
A Non-Data-AidedMaximum
Likelihood Time DelayEstimator using
ImportanceSampling
'A Non-Data-AidedMaximum Li'A. Masmoudi,F. Bellili, S. Affes, andA. St6phenne,
kelihood Time Delay EstimatorUsing ImportanceSampling",IEEE Trans.on Sign.Process.,vol. 59, no. 10,pp. 4505-4515,October
2011.
DOI: 10.1109/TSP.2011.2161293
IEEE (c)
Abstract
Danscet article,nouspr6sentonsun nouvelestimateurir maximum de vraisemblancepour
le retard de propagationbas6sur "importancesampling" (IS). Nous montronsque la rechercheexhaustiveet les probldmesde convergencedont souffrentles m6thodesit6ratives
peuvent6tre contourn6s.Les donn6estransmisessont suppos6sinconnues.Le retardreste
constantsur l'intervalled'observationet le signalestentach6par un bruit blancgaussien.
NousutilisonsIS pour trouverle maximumglobalde la fonctionde vraisemblance.
L id6e
du nouvel estimateurest de g€n€rerdesr6alisationsi partir d'une versionsimplifi6ede la
fonction de vraisemblance.Nous verronsque les parambtresde l'algorithme affectentles
performancesd'estimationet qu'avecun choix appropri6de cesparamdtres,le retardpeut
Otreestim6de faqonpr6cise.
In this paper,we presenta new time delay maximum likelihood estimatorbasedon importancesampling(IS). We show that a grid searchand lack of convergencefrom which
most iterativeestimatorssuffercan be avoided.It is assumedthat the transmitteddataare
completelyunknown at the receiver.Moreoverthe carrier phaseis consideredas an unknown nuisanceparameter.The time delay remainsconstantover the observationinterval
and the receivedsignal is corruptedby additive white Gaussiannoise (AWGN). We use
importancesamplingto find the globalmaximumof the compressed
likelihoodfunction.
Basedon a global optimizationprocedure,the main ideaof the new estimatoris to generate
realizationsof a random variableusing an importancefunction, which approximatesthe
actualcompressedlikelihood function.We will seethat the algorithmparametersaffectthe
estimationperformanceand that with an appropriateparameterchoice,evenover a small
observationinterval,the time delaycanbe accuratelyestimatedat far lower computational
costthan with classicaliterativemethods.
2.1 Introduction
Parameterestimationis a crucial operationfor any digital receiver; in particularthe recovery of time delay introducedby the channel.Typically, in network communications,the
time delayis usuallyassumedto be confinedwithin the symbolduration[1]. Particularly,
symbol timing recoveryallows for samplingthe signal at accuratetime instantsin order
to achievesatisfactoryperformances.The key task of timing recoveryconsistsin determining the time instantsat which the receivedsignal shouldbe sampledin order to perform
reliabledatarecovery.However,in many other applicationssuchasradaror sonarsystems
l2l l3l, whereit can exceedthe symbol duration,the time delay is usedto localizetargets.
During the last few decades,many time-delayestimatorshave been developedtrying to
achievethe well-known Cram6r-Raolower bound (CRLB). A key step in time recovery
16
schemesis the determinationof an objectivefunction from the statisticsof the received
signalfrom which an estimateof the time delay canbe extracted.In this sense,it is known
that the maximum likelihood estimatoris an asymptoticallyefficient estimator,and that it
performscloseto the CRLB at relativelyhigh SNR values[7], evenfor shortdatarecords.
Therefore,it hasbeensubjectto intenseresearch.In the caseof data-aided
transmissions,
where the transmitteddata arc a pri,ori completely known, an expressionfor the global maximum of the log-likelihood function is analytically tractable.However,when the
transmitteddataare completelyunknown (i.e., the parameterof interestshouldbe blindly
estimated),the log-likelihood function becomesextremelynon-linearand it is difficult to
analyticallyfind its global maximum.In this case,maximumlikelihood (ML) solutions
must be numerically tackled. The grid searchtechniqueis the most basic alternativeto
numericallylind the maximum of the non-linearlikelihood function. Unfortunately,this
techniquecan be usedonly if the range of the parameteris confinedto a finite interval,
otherwise,iterativemaximizationproceduresmust be envisaged.The most famousiterative proceduresare the Newton-Raphsonmethod [5] and the expectation-maximization
algorithm [6]. However,thesetwo prominentmethodsare known to convergeto the ML
solution only if the initial guessis close enoughto the true unknown parametervalue.If
not, theseiterative algorithmsmay convergeto a local maximum of the likelihood function, or evendiverge.To circumventthis problem,thesealgorithmsmay usemany initial
valuesto improvetheir performance.But this increasesin counterparttheir computational
complexity without evenultimately warrantingtheir convergenceto the global maximum.
In this work, we resortto an entirelydifferent approachfor the estimationof the time delay
parameter.The compressed
likelihood function is derivedconsideringthe transmittedsymbols as unknown but deterministic.Basedon this function, an iterative algorithm earlier
implementedin [8] performsbetter in the high SNR region than the low-SNR unconditional ML (UML) timing error detectors(TEDs) [1], but its performancestill dependson
the initialization value making it thereforeprone to severedegradationdue convergence
uncertainty.
Motivatedby thesefacts,we developin this papera new non-iterativeapproachto find the
time delay conditional maximum likelihood (CML) estimates.We implement the CML
algorithmin a non-iterativeway.We avoid the grid search,essentialin traditionaliterative
approaches,by using the importancesamplingtechniquewhich has been shown to be a
powerful tool in performing NDA ML estimation.In fact, this method was successfully
applied to estimateother crucial parameterssuch as the direction of arrival (DOA) [9],
the carrierfrequency[10] or thejoint DOA-Dopplerfrequencytl11. The importancesampling techniqueis used in this paperin the context of time delay estimation.Moreover,
we adoptthe discrete-time
modelwidely usedin the field of sensorsarrayprocessing[12]
and more recently formulatedin the context of time-delayestimation[8]. The resulting
IS-basedestimatorattainsthe modifiedCRLB (MCRLB) overboth the mediumandhieh
t7
SNR regions,whereasthe traditional UML TED, being derivedunder the assumptionof
low SNR, doesnot approachthe MCRLB at the high SNR region.
The remainderof this paperis organizedas follows. In sectionII, we presentthe discretetime signalmodel that will be usedthroughoutthis article.We derivethe compressedlikelihoodfunctionin sectionIII.In sectionIV we introducetheimportancesamplingmethod
that will be usedin this article to find the global maximum of the compressedlikelihood
function. SectionV dealswith the choiceof the importancefunction and discussesthe impact of someparameterson the estimatorperformance.The newly proposedalgorithm is
developedin sectionVL Simulationresultsarediscussed
in sectionVII and,finally,some
concludinsremarksare drawn out in sectionVIIL
2.2 Discrete-TimeSignal Model
First,we presenta list of notationsanddefinitionsthat will be usedin this article.
E.{}:
theexpectation
with respecttor.
Euclideannorm.
ll Il ,
(.)t, (.)t : transpositionandconjugatetransposition.
SNR :
signalto noiseratio.
IS :
importancesampling.
ML :
maximum likelihood.
CML :
conditionalmaximum likelihood.
MCRLB :
Modified Cram6r-Raolower bound.
QAM :
PAM :
quadratureamplitudemodulation.
pulse-amplitudemodulation.
Considera traditional communicationsystemwhere on one hand the channeldelaysthe
transmittedsignal and on the other hand an AWGN with an overall power of l/6 corrupts
the receivedsignalas follows :
a(t) : t/ E, r(t - r*)eio+ w(t),
(2.r)
where r* is the unknown time delay to be estimated,0is the unknown but deterministic
channeldistortionphase,tu(f) is an additivewhite Gaussiannoise (AWGN) with independentreal andimaginaryparts,eachof varianceNsl2 andt/4
is the signalamplitude.
The unknowntransmittedsignalr(f ) is modeledasfollows :
n-r
,,\
r(t):)
\a-
q h,(/t,- i T ) .
/J
(2.2)
i:o
whereK is the numberof transmittedsymbolsin the observationinterval, {"i}fJ'are the
unknown complex-valuedsymbols,h(f ) is the shapingpulse of energy E6 and7 is the
18
symbol'sduration.
In the sequel,we outlinethe discrete-timesignalmodel which was proposedfor the lirst
time in [8] to derivean iterativeCML timing recoveryalgorithm.The receivedsignaly(f)
is passedthrough an ideal lowpassfilter of bandwidth F"f 2 and sampledat a frequency
F" : 7lT" : klT, where k is a given integer which guaranteesthat fl" is abovethe
Nyquistrate.Then,the receivedsamplesg : la(O),A(7"), U(27"),...,y((M - 1)7")l'
canbe written in a matrix form as follows :
(2.3)
A:A,-rlus,
whereM is the numberof samplesof g(t) andu andA"- are definedas follows :
6 : [ ' u ( 0 ) ,w ( 7 " ) , . . . , u ( ( M - 1 ) ? : ) ] t ,
A,* - [oo(r-), at(r*), ..., ax-r(r*)],
(2.4)
(2.s)
with
a t ( r * ) : l h ( - i T - r * ) , h ( 7 " - d T - r * ) , . . . , h ( ( M - 7 ) 7 "- i T - , . ) ] ' .
(2.6)
In(2.3), r is the setof unknowndataand signalphasewhich is givenby :
r : ceio : lco, c1 ... c6-1fT eie.
(2.7)
Moreover,the covariancematrix of u.ris given by :
C* : 2o2Ix4 :
N6F"I7,1,
(2.8)
where-Iy refersto the (M x M) identity matrix and2o2: l/6F". The sampleddatagris a
linear function of the vectorr but dependsnon-linearlyon the time delay r*. We mention
in [8] is inspiredfrom themodelwidely usedin array
thatthe modelof Eq. (2.3)presented
signal processingwhere each column of the transfer matrix is a function of a different
parameter,usuallythe direction-of-arrivalor the frequencyof eachincoming signal.In the
contextof time delay estimation,the entirematrix -4". dependson the sameparameterr* .
2.3 Likelihood function
The conditional likelihood function of the observed data g is given by :
t v ( s l * ; rx) p ( y l r ; r ) : C e x p
{- #
l l a- A , * l l ' } ,
Q.s)
is the probabilitydensityfunction (pdfl of y conditionedon r and parameterizedby r, and C is a positive constantwhich doesnot dependon the time delay
wherep(Alr;r)
T9
and thereforewill be dropped,without loss of generality.Note here that r is any possible
valueof the time delayparameterr" and that |v(Al*;r) attainsits maximumat r : 7*,
i.e.,r* - argmaxA(grlr;r).
T
Actually,one needsto maximize |t(gln;r) with respectto r in orderto find the ML solution F. However,(2.9) imposesa joint estimationof c and r*, which is very difficult
to perform.Therefore,two principal approachesare developedin the literaturein order to
obtain a likelihood function that dependsonly on r. On one side,the unconditionalmaximum likelihood (UML) estimatorintroducedin [] considersthe datasymbolsas random
and henceaveragesthejoint likelihood function over re to obtain a function that depends
only on the time delay as follows :
A(sl'): n,UY(ul*;r)j.
( 2.10)
On the other side,the data symbolsare modeledas unknownbut deterministicin the formulation of the conditionallikelihood function. Therefore.8. the solutionthat maximizes
(2.9)with respectto r,for a givenr is usedin(2.9) asa substituteof r. Actually,i which
maximizeslv(A;r,r) alsomaximizesthe log-likelihoodfunctiongivenby :
L(u;*,r): -:2 o 2 "ll" a - A,* ll'.
Therefore,takingthe gradientof L(ylr;z)
(2.rr)
with respectto aoandsettingitto zero'.
oL(a_;r,r)
: _4toTa_ ATA,*):0,
0ro
(2.r2)
yields the following result :
a : (ATA")-'ATy
: ATa,
(2.r3)
whereAf, : (AT A")-r AT is the pseudo-inverse
of the matrix A". SubstitutingO into
(2.I1), one obtainsthe so-calledcompressedlikelihood function, that dependsonly on the
unknowntime delayparameter:
L ( a ; r , O: - : - a H ( I x - A , A f ) a ,
zo'
(2.r4)
which can be further simplified by dropping the constant terms to obtain the useful compressedlikelihood function denotedby L.(a; r) as follows :
L"(a;i : sHA,(ATn,)-' nTu.
(2.r5)
Note that the expressionin (2.I5) representsthe cross-energybetweenthe pseudo-inverse
filter Af and the sampledmatchedfilter Al . For z equal to the timing parameterto be
estimated,the fllter Af becomesa zero-forcingequalizersincethe componentsof Af A
areintersymbolinterference(ISI)-free(i.e.,Af A : n -l Af, -, see[8]).
20
2.4 Global Maximization of the CompressedLikelihood
Function
To perform maximum likelihood estimation,we have to maximize (2.I5) with respectto
r. Unfortunately,a closed-formexpressionfor this optimization problem is not analytically tractablesincethe objectivefunctionin (2.15)is extremelynon-linearwith respect
to r. Therefore,many methodshavebeendevelopedto numericallyfind the maximum,
but most of them are iterative [-8]. We cannot deny that thesemethodsprovide good
performancein terms of error variance,but unfortunatelythey require, in counterpart,a
sufficiently closeinitial guessto convergeto the global maximum of the likelihood function. Otherwise,the resultmay be a local maximum,which doesnot correspondto the true
time delayvalue.This is why a suboptimalalgorithmneedsto be appliedfirstly andthen
its output is consideredas an initial valuefor any iterativetechnique.
To avoid this challengingdrawbackof iterativemethods,we proposein this paperan entirely differenttechniquewhich doesnot claim any initial guessof the time delayparameter.
We apply the global maximizationmethodearlierproposedby Pincus[13] which provides
a powerful tool for accomplishingnonlinearoptimizationand guaranteesfinding the global maximumwithout any initialization concerns.In fact, the theoremof Pincusstatesthat
themaximumofL.(g;r) is givenby :
e:lim
frrL'",r(r)d,r,
(2.16)
where
r|
l - \ --
"c,p\t)
exp{pL.(s;r)}
(
f
r
/
\l
r
|,exDl OL-tutrttar
)
(2.r7)
JJ
can be viewedas the normalizedfunctionof exp{pL"(A;r)}.Note that in (2.16)and
(2.17),.I is the integrationintervalin which r is supposedto be confined.In a certain
way,L'",,(r) canbe viewedas a pdf (sinceit verifiesall the propertiesof a pdf), but since
r is actuallydeterministic,L'",r(r) is moreconvenientlycalleda pseudo-pdf[9]. It is also
worth noting that, as p -+ oo, L",r(r) becomesa Dirac delta function centeredat the
location of its original maximum.We leavebroaddetailson this point in Appendix A.
The ML estimatorfor the time delay parameter,obtainedfrom the location of the global
maximumof L"(y;r) is given,for a largevalueof po,by:
,:
rL'",ooQ)dr.
(2.18)
/,
Now, we needto evaluatethe integralgivenin (2.18),althougha direct integrationremains
alwaysdifficult if not impossible.However,this integralis in a way the meanvalueof a
randomvariabledistributedaccordingto L'",ro(.).It was shownin [1a] that this type of
2I
integralcanbe efficientlyevaluatedusingMonte-Carlosimulationsasfollows :
;.: +f,r,
(2.re)
arerealizationsof r distributedaccordingto the pseudo-pdf,L'",ro(r), and
hence the global maximization problem reducessimply to a generationof random variables. Yet, since it is a non-linear function of r, the direct generationof realizations
where {rr}t:,
accordingto L'",00(r)is computationallyhard.Thus,insteadof pursuinga fruitlesspath,
we usethe importancesamplingtechnique,as donein [9], [10] and [11] for the estimation of the signaldirectionsof arrival, the carrier frequencyand the Doppler frequencies,
insteadof directlyusing(2.17).
2.5 The Importance Sampling Technique
lt has been shownthat the importancesamplingtechniqueis a powerful tool to compute
multipleintegrals;in particularthe one given in (2.18).In fact, it can be easilyseenthat
for anyfunction2/(.) :
: rr,lffi s'(r)d,r,
l,
f, f {r)r'",,,e)d,r
(2.20)
whereg'(.), calledthe normalizedimportancefunction,is anotherpseudo-pdfwhich must
be chosenas a simplefunctionof r so thatrealizationsdistributedaccordingto g'(.) can
be easily generated.Then, the Monte-Carlo method is used to empirically computethe
integralin (2.20) simply via the following summation:
-.1L'"'oo('r)
- 1 + pt 1
'rt
?)t'",'o?)d,r
/
2
t
g,0i
R
lrf
(2.2r)
wheren, is the kthrealization of r accordingto the normalizedimportancefunction g'(.)
Typicully,g'(.) andL'",00(.)shouldbe very similarto
and E is the numberof realizations.
reducethe varianceof the estimates.However,L'",ro(.)remainsa complexfunction and
in counterpartS'(.) needsto be as simpleas possible.Therefore,sometrade-offsmustbe
found in the constructionof the importancefunction.In fact, the inversematrix (AT A")-t
in the actualcompressed
likelihood function,L"(A;r) (or equivalentlyL'",r.(.)), is very
non-linearwith respectto r. Intuitively, onecanreplacethis inversematrix by the diagonal
matrix ht*
Hence,a reasonableapproximationof the compressedlikelihood function
is:
=
L"(a;r)
fiu',+"ATa.
2In our case,we have
f(r) : r.
22
(2.22)
t
The approximationof (Al A")-1 with ftt* is very reasonable
for mostof theconventional pulseshapingfunctions.For instance,it canbe veriliedthat for the widely usedsquare
root-raisedcosinepulse,the diagonalelementsof (Al A,)-1 are dominantcomparedto
its off-diagonalones.In fact,as definedin (2.5),the columnsof A" arebuilt upon shifted
versionsof the shapingpulse h (.) , thereforeevery elementof AT A" can be seenas the
convolutionof two shiftedversionsof h ( .) (the shift being an integermultiple of 7), which
valueis maximumwhenthe shiftis the same,i.e.,in thediagonalelements.
Whereas,when
the shift is not the same,the valueof the convolutionis very low. SeeAppendix B for more
detailsaboutthis observation.In the particularcasewherethe pulse shapedoesnot generateinter-symbolinterference,the approximationbecomesstrict equalityand (2.22)yields
the exactcompressedlikelihood function. Then,a reasonableimportancefunction is given
by:
gr r ( , )
(
K-r
\
k:0
r lM-r
- exp o'
{| f- I l s l # " l tlI: 0 y"(i,r")h(i,r"kr
.,1'),
( 2.23)
wherep1 is anotherconstantdifferent3from p6. Note that the normalizationof go,(.) by
It gor@)d, yields the normalizedimportancefunctiong'or(.){i.e., 9'r,(r) : ##W).
But sincethe periodogramof the dataevaluatedat the time delayr, In(r), is givenby :
:
16(r)
lM-|
l'
:0, 1'2,"',K -r,
lDr.(u'")h(ir"-kr- ')1,
(2.24)
l;:o
I
then,we rewrite the importancefunction as follows :
n-l
oo',?):fl e"p{pilt(')},
k:0
(2.2s)
with
or:fin,
Q.26)
The normafizatronot(2.25)leadsto the pseudo-pdfg'(.) which will be used,hereafter,to
generatethe realizationsinvolvedin (21) :
K-l
l{ exp{p'r16(r)}
nt.l-\vp\\' / -
a:0
(2.27)
I,E
exp{p'rlp(u)}du
lt is also worth noting that the performanceof the new maximum-likelihoodestimator
rln our case,p1 can be equalto ps, unlike for the multiple parametersestimationwhere p1 should be
different from po.
23
dependson thechoiceof p\.ln fact,our ultimategoalis to find the globalmaximumof the
functionL.(g;r) : gH A"(ATA")-'ATg.
However,this function exhibitsmany local
maxima evenin the total absenceof noise,and it is often difficult to distinguishbetween
the global and a local maximum. For this pu{pose,p', is chosento render the objective
function in (2.27) more peakedaroundits global maximum which will have a relatively
higherpeakcomparedto the local maxima.This behavioris illustratedin Fig. 2.1, which
plotsthe functiong'oi(.)for Pl: L0 andp't : 20,inthe total absence
of the additivenoise.
Moreover,we showin Appendix C how this parameterrendersg'o,"(.)morepeakedaround
its global maximum.
--T-
(
r'+27
T
,l
a-
,t
FtcuRe 2.1- Plot of g'rl(.) for p'r: 20 andp\ : L0 usinga root-raised
cosinepulseand
f o rK :
100.
Basedon this fact, it can be stated,a priori, that it is betterto arbitrarily increasep', in
orderto achievebetterperformance.But, this is much easiersaidthan donesince,in practice,this leadsto numericaloverflows.Actually,the bestvalueof p', is the highestpossible
withoutresultingin any overflowin the computationof g'0,(.) as it will be seenin Section
VII. Sameargumentis valid for ps.In fact, the approximationof (AlA,)-'Vy
meansthat the quantiryL"(U;i : .ynA,(AT A")-t ATg is almostequalto
h.l*
fifP 1r1
wherefYQ) : gHA,Aly (notethatthe superscript(a) in fL:)1r) refersto the word
"approximate"wherewe approximateL"(g;z) with LY)(r) by replacing(Al A")-r by
#t*).
This meansthat ps will resultsin an overflowin exp{p6L.(y;r)} as far as p1
24
Therefore,the optimalvalue,p|t , of po ve{otf;fL^ fr;}.
rifiesp'{t : p?'where p?' is againthe optimalvalueof Or: frp',,..
resultsin an overflowin exp
We alsonotethat the matrix A, in(2.22) exhibitsan interestingstructuresinceits columns
aresimply shiftedversionsof the pulseshape.Hence,the matrix-by-vectoroperationAly
can be viewedas a filtering operationof the receivedsamplesgrwith a filter h(.) whose
coefficientsarethe centralrow of Af . Therefore,the computationof (2.27)is quite simple
and realizationsdistributedaccordingto S'oi(.)in (2.27)can be easiergeneratedthan accordingto Lt.,oo(.)
in(2.17).
2,6 Estimationof the Time Delav
In the sequel,two scenarioswill be proposeddependingon the rangeof the unknowntime
delayparameter.In the first scenario,we assumethat the time delayparametertakesvalues
within 10,PT), whereP is a strictlypositiveinteger(i.e.,P > 1). In the secondscenario,
we assumethat the time delayparametertakesvalueswithin [0,7].
2.6.L First scenario2r* e 10,PTI
As already mentionedin the introduction, in many applicationssuch as radar or sonar
transmissions,the actualtime delay introducedby the channelmay exceedthe symbol's
duration.In this subsection,we assumehoweverthat the time delay doesnot exceedPT,
where P is a given strictly positiveinteger,i.e., r* € 10,PT| This upperlimitation of the
intervalis justified since,in eachcommunicationsystem,we alwayshavean a pri,ori, rdea
aboutthe maximum rangeaof r. As we have seenin the previoussection,the maxima of
g'r,r(.)areperiodic, with period 7. Therefore,many secondarypeaksmay appearwhich
of r*. In fact, to obtain unbiasedestimatesof r*, the
expectedvalueof the estimationerror re : T* - i shouldbe equalto zero,i.e. :
ultimately affectsthe estimatei
(2.28)
E{r"}:E{r*-i}:0.
However,it may occur that the differencebetweenr* and z* is very important.In fact, to
simplify, assumethat gtr,(.) has only 2 peaksand neglectthe others.Then the generated
valueswill takevaluesaroundr* andT + T*, with higher probability aroundr* wherethe
highestpeak is located.If we denoteby Cl the set of realizationstaking valuesnear r*
and Cz the set of realizationstaking valuesnearr* * 7, then from (2.21) the estimated,
?, canbeapproximatedby :
^
'
-_
card(Ct) _* , card(Cz) /_*
I r-61
\/
R
.rr\
rt),
4Notethat P can be alwayschosenas large as desiredto ensurethat r* €
10,PTI
25
(2.2e)
with card(C)denotingthecardinalof C, i.e.,thenumberof elementsof C. Thereforer* f
r* since R : card(C) + card(C2)and card(C2) is alwaysnot equalto zero.Moreover,
the bias is largerat low SNRsand/orshortdatarecords.This propertywas alsopreviously
observedin the caseof frequencyestimationin [15].
To circumventthis problem,the pseudo-pdf,g'r,r(.),must be centeredaroundr*. To that
end, two intuitive methodsmay be envisioned.We may either eliminate the secondary
peaksto keep only the principal one, or we can generateother peaksin a way that the
numberof secondarylobes on either side of the principal lobe is the same.The first idea
seemsto be the most efficient, but it is unfortunatelyunrealizableand we opt for the
secondalternative.Indeed, as we have seen,the estimationbias stemsfrom the peaks
taking place after the principal lobe. Thus we have to modify fii(.) so that it becomes
quasi-symmetricaroundr*. To that end, the simplestway is to suppose,virtually, that r
takesnegativevaluesalthoughr is always positive.We extendthe interval of definition
of g'p\(.)from [0,P7] to [-QT,P7], where Q is a positiveintegersmallerthan P. In
this way, virtual secondarylobes appearbefore as well as after r*. Moreover,as it can be
seenfrom Frg.2.2, the probability of generatingrealizationsaroundr* I iT is almostthe
sameas the one of generatingrealizationsaroundr* - iT . Hence,the estimatorbecomes
unbiasedand the estimatei
o
is more accurate.So far, we have establishedan unbiased
0
r
o
o
o.4
o
i
,a +T
:.f
f
T
0.5
:.I
1
711
Frcune 2.2-Plotof thecumulativedistributionfunction(CDF),G'(r) whosepdf is g'(r),
SNR = 5 dB.
estimatorbasedon a linear averageof the generatedrealizations.But throughsimulations,
we noteda performancechangeaccordingto the constellationtype. In fact, for a constant-
26
envelopeconstellationsuch as phase-shiftkeying (PSK), the estimatorworks perfectly.
However,its performancedegradesdramaticallyfor non-constant-modulus
constellations
such as pulse-amplitudemodulation (PAM), quadratureamplitude modulation (QAM),
etc. In fact, aswe havepreviouslyseen,the main problemthat facesthe new ML estimator
is the presenceof secondarypeaks.Although we have explained,in sectionV how to
reducethe adverseeffectsof thesepeaks,they generateirreversibleerrorsin the caseof
non-constant-envelope
constellations.To simplify the problem,without loss of generality,
we supposeagainthat gii(.) exhiUitsonly two peaks,one located atr* and the other
locatedat r* 17. From (2.25),gp,,(r-) andgr,,(r. -t T) canbe written asfollows :
(
K-tlM-L
- kr gp,r(r*):""p1
y"(r")h(ir"
pi t- * : r lt
l-o--o
(
l')
(
l*-t
| )
\
ri:o
-r
r-)l | *o{ piIt y.(ir")h(ir"
.,1')
,
(2.30)
and
K-1.lM-r
gri|. *T)
:
*,
{rt
*o
_kr_-.,1,)
,
tz-.r lt
a.GT")h(iT"
'
lz-t "
k:r
I i:0
- Kr-".,1')
{,r lTnr*")h('ir"
Q.3I)
Noting that the samplesof the receivedsignalarelimited in time to 10,KTI andthe magnitudesof h(t - Kf - r) for , € [0,KTI arevery smallcomparedto thoseof h(t - KT - r)
y. (i,T")h,(ir"- KT - r) canbeneglected.
for t € lKT, (K + 1)7], then,the term !f;1
Consideringthis result,we expressgpt(r*) as a functionof goiQ. + T) :
gri?.) = gp,r(r*+ 7) exp{p'rlo(r*)},
(2.32)
where 1o(".) dependsmainly on the amplitudeof the first symbol.In practice,it may
occurthat the amplitudeof the first transmittedsymbolis the smallestone.In this case,
the contributionof exp{pllo(r.)} in goi?.) is far lessimportantthanthe otherterms,i.e.,
e x p { p | l s ( r . ) }< e x p { p l l i ( r - ) } f o r i , : 1 , 2 , . . . , K - T . A s a r e s u l t , g o i ( r . ) w i l l b e
closerto go,r(r*lT), whichis a localmaximummakingtheestimatei shifttowardr* *7.
The sameproblem occurswhen the last transmittedsymbol has the smallestamplitude
with the only differencethat the shift will be toward r* - T.
To avoid theseproblems,16(r*) must be as large as possible.To that end, we slightly
modify the algorithm in the caseof non-constant-modulus
constellationsby sendingtwo
a priori known symbols: one at the beginningand one at the end of the frame.Moreover,
thesetwo known symbolsmustbe of highestenergyamongthe constellationpoints.Then,
27
1o(r.) is no longernegligiblecomparedto li(r*), for i, I 0, and the differencebetween
the magnitudeof gr, (r.) andgpi(r. + 7) is largeenoughto avoidan importantdetection
error.The samething holdsfor got(r*) andgo,?- - T).
2.6.2 Secondscenarioi,r* € 10,7]
In many cases,the time delay doesnot exceedthe symbol duration7. Therefore,we must
look for the global maximumonly in [0,7]. As previouslyexplained,the maximaof the
importancefunction areperiodicallylocated,with a period equalto 7. Moreover,sincewe
know a priori that r doesnot exceed7, then we can more convenientlyusethe circulat'
(insteadof the linear) meanto evaluatethe meanin (18). It will be seenin sectionVII
in the
that the use of the circular meanprovidesconsiderableperformanceenhancements
low-SNR region. As it will be explainedlater, the use of the circular meanconsiderably
reducesthe computationalcost.
To introducethe conceptof a circular mean,considera circular random variable which
takesvaluesin a finite interval that can be mappedinto the unit circle. For instance,let
a be a randomvariabledefinedin [0, 1] with pdf P(").Then, the circularmeanof a is
definedas :
E.{o} :
}z
I
P (a)cl,a,
e>tp{jura}
(2.33)
whereZ denotesthe anglein radians.Having Rrealizationsof o, its circular meanis [6] :
^: *t**"'ornva.)
(2.34)
In our case,if thetime delayis not confinedwithin theinterval10,1],it canbe easilytransposedinto this intervalby normalizingr* by 7. Then,the resultingtransposedestimateis
inversedto obtain an estimatein the orieinal interval. Hence.the IS estimateof r* usins
(2.34)and(2.21)is :
*:#'*f='*",{t!},
(2.3s)
,: #t**on;"o{iT},
(2.36)
or finally :
where
F(r):W
(2.37)
9 oi\r )
5Notethat the circular mean cannotbe usedin the first scenariowhen r may exceedT since it always
returnsan estimatein [0,
by virtually bringing, into this interval,all the secondarylobesof the normalized
"]
importancefunction.
28
Note that we needto find the angleof a complexnumber,and thus,we can removeany
positivereal factor taking placein (2,36)without affectingthe linal result.This meansthat
the two strictlypositivenormalizationconstants[, f'@)au and[, g'(u)du canbe simply
dropped.Moreover,an overflowmay occur sinceboth the numeratorandthe denominator
are exponentials.To circumventthis problem,we replaceup (r) by F'(16) :
K-'
(
}{-l
(rot"(',) -d,D,-(',))
- drU, tt(rr)-,-g
F'(rt)- exp
lro;{r,)
k:0
\
k:0
(2.38)
)
wherewe multiply F(r) by a real scalarfactor.
2.6.3 Summary of steps
In the following, we summarizethe stepsof the new algorithm for the two considered
scenarios.
1 . Basedon the sampleddatag(i,T),'i : 0, I,. . ., M - T,evaluate
theperiodogram
Ia(r) accordingto (2.24).
2 . Computethe normalizedimportancefunction in (2.27). Note that, in practice,we
use a discreetmodel by substitutingthe integrationin the denominatorof (2.27) by
a summationas follows :
n-l
(
T-T
t
r
/
\)
| | e x D {o . l u l r l f
t-r
k:o
-nt . (-\
vplv t
n-l
i
(2.3e)
D):' fl exp{pil*(',)}
,k:0
wherel/ is the total numberof points in the time delay interval.
3 . GenerateR realizationsof the parameter{r}L,
using the inverseprobability inte-
gration as detailedin AppendixD.
4 . EvaluatetheweightcoefficientF(4) delinedin (2.37)(or F'(r) definedin (2.38)if
we considerthatr is in [0,7]) for eachgenerated
valuer;.
5. Computethe meanof the generatedvariablesmultiplied by the weight coefficients
to find the ML estimateof the time delav.
oNotethat the samesimplificationshavebeenusedin
[9] to estimatethe signalDOA.
29
2.7 Simulation Results
In this section,we will presentnumericalresultsto substantiatethe performanceof the
new ML estimatoras a function of the SNR. We will also refer to our new IS-basedML
estimatoras "IS algorithm". The normalized(by T2) meansquareerror (NMSE), defined
in (2.40),will be usedas our performancemeasure:
NMSE(r) :
E{(} - r-)'}
(2.40)
ry1)
1-
andcomputedover 1000Monte-Carloruns.The modifiedCram6r-Raolower bound(MCRLB)
is also normalizedby T' and the total numberof transmittedsymbols,K, in the observation window is set to K : 100 and p', is taken equal to 28. Unless specifiedotherwise,
a root-raisedcosineshapingpulseof roll-off factor of 0.5 is used.First, the effect of p1
(or equivalentlypt) on the performanceof our IS-basedML estimatoris shown in Fig.
2.3 at an SNR of 10 dB. As it could be predicted,the meansquareerror decreases
as p1
increasestoward its optimal value and, for too large values,the performancedeteriorates
due to numericaloverflows.In the implementation,p', can be set as a function of the po-
too
' ' : ' ' S N R+
to-'
10dB
:
:::
uJ.
o 10'
=-
10
-
'''N::',
:,,'l
1 o -'t0
'L
P't
versusp\for SNR=10dB.
FtcuRB 2.3 - Performance
wer of the receivedsamplesg. Moreover,using computersimulations,we verify that for
a root-raisedcosinefilter the ratioL|'o?) I g'O) is almostequalto 1. Then to reducethe
computationalcomplexity,we can setthis ratio to 1 in (2.2I) and (2.36).In fact, Fig.2.4
30
showsthe NMSE of the IS-basedtime delay ML estimatorwhen this ratio is preserved
in the importancefunction and when it is set to 1. As it can be seen,this simplilication
doesnot degradethe performanceof the estimatorwhile reducingthe computationalcomplexity considerably.This simplification is also valid for any linear modulation scheme.
Therefore,in the following simulationswe considerthat :
r|
"c,p6\'
(-\
,/\
I o'r\r )
)
.
(2.4r)
Note that this simplificationremainsvalid when the intersymbolinterferenceis not important (for high valuesof the roll-off factor). As the roll-off factor tendsto 0, it appears
necessaryto considerthe ratio in order to achievebetter performanceof the estimator.
Moreover,we implementthe iterativeCML estimator,called CML-TED, proposedin [8]
o
z
10
SNR dB
FtcunB 2.4
Estimation performance considering L'",roQ)f g'(r)
and setting
LL,^?) ls'(r) to 1 wirh QPSKmodulation.
and compareits performanceto the performanceof our IS-basedCML estimator.As far
as we know, amongall the existing synchronizationtechniques,the CML-TED algorithm
achievesthe best performance,but, as an iterative procedure,its performancedepends
strictly on the initial guess.To corroborateour claims, we considerin Fig. 5 two initial
valuesof r* for the CML-TED, which shouldbe seenasthe result of anotherestimator.In
Ft5.2.5,the smallcrossesrepresent
the normalizedvariancewheretheinitial valueis very
closeto thetruetime delayvalue,i.e.,verifies116-r*l : T 170,with r* beingthetruetime
delayvalueto be estimatedand rj is the initial guess.As it canbe seenfrom this figure,
31
evenwith a close-enoughinitial guess,our IS-basedestimatoroutperformsthe CML-TED
estimatorin the high SNR region althoughthe CML-TED achievesbetterperformanceat
low SNR values.We also note that, at high SNR values,the performanceof our IS-based
estimatoris closeto the MCRLB. This meansthat, in this region,our new time delay estimator exhibits performancesequivalentto thosethat could be achievedif the transmitted
data were perfectly known to the receiver.However,if we considerlrj - r* | : T f 2, the
performanceof the CML-TED deterioratesconsiderablyover the entire SNR region.This
illustratesthe fact that the CML-TED algorithm fails to estimatethe time delay if the initial value is not appropriatelychosen,while no initialization concernsare raisedwith our
new IS-basedCML estimator.Moreover,the secondvariant of the IS algorithm, namely
o
o
z
'L
10
-5
10
SNR dB
Ftcune 2.5 - Comparisonbetweenthe estimationperformanceof the IS algorithm using
the two scenariosandthe trackingperformanceof the CML-TED usingQPSK modulation.
consideringthe time delay as a circularvariable,is also represented
in Fig. 2.5. We see
in this casethat the varianceeffor is reducedin the low SNR region. In addition,in both
cases,starting from an SNR value of about 5 dB, our IS-basedalgorithm surpassesthe
iterativealgorithm,evenwhen assuminga sufficientlyaccurateinitial guess.
Furthermore,in Fig. 2.6, the CML-TED algorithm exhibits a variancepenalty for a
roll-off equal to 0.2. This penaltyis higher for smallerexcessbandwidth.It has been
shownin [8] that the CML-TED reachesthe asymptoticcompressedCRLB (CRLB"),
and the differencebetweenthe MCRLB and the CRLB" becomesmore important as the
roll-offfactor decreases.
Then the performanceof the CML-TED cannotapproachasympJL
totically the MCRLB for small roll-off factors.In contrast,the new IS-basedalgorithm
always reachesthe MCRLB in the high SNR range,irrespectivelyof the roll-off factor
value.
:
i:
:o-: O :
:
'''''''''''''\'
:
l - - O- - r s a r s o
, . . , , , , , , , ,I,
: : , : , : , , , , , , , , 1 O ' , , , , , : : : : : : : : : , ' ' |' -' .- :r - " " . r . o
. . . " e . . . . . . . . . ......' .i . . .. . .. : . . . . .. . I
ri,
o
t0
d
..t
.,.
,..... . -MCRLB
I
1t
,::,::
: : . : : : : : : : : : : : . \ : : : : : : i i : i : : : : : : : :: : :
: : : : : : : : : . : . : . ' :: i:{: ' : : :: i : : : : : : : : : : : . .
..:.::::.
.:.:
{ : : : : : : . . . . . . . . . . . . t i : : 1 . . : : : : : .: '
::::::::::::::::::::::::
. : : : : : : : : . :: :: : : : : : : : : :
. ::::::.......:...:
o
10o
I
z
10
SNR dB
FIcURB 2.6 - Comparisonbetweenthe estimationperformanceof IS algorithm and the
tracking performanceof CML TED using QPSK modulation and for a roll-off factor of
0.2.
In Fig. 2.7, performancecurves are drawn for 16-QAM and 64-QAM' as examplesfor
non-constant
modulusconstellations.
As explainedin sectionVI, we force the first and
the last transmittedsymbolsto be of maximum energy.To illustratethe performancedegradationin the caseof higher-ordermodulations,we alsoplot the NMSE for QPSK.As
we can see,the IS algorithm achievescloseperformancefor the threemodulationsordets,
with, however,a small improvementfor the QPSK modulation.To illustrate the performanceenhancementachievedby forcing the f,rst and the last symbolsto havemaximum
constellationmagnitude,we plot, in Fig. 2.8 the performanceof the new IS-basedML
estimatorwithout this constraint.As anticipated,the performanceis strongly affectedby
the two edgesymbolssincethe curve correspondingto the non-forcedsymbolsdoesnot
approachthe MCRLB. Therefore,for non-constant-envelope
modulatedconstellations,it
is essentialto force the first and the last transmittedsymbolsto havemaximum energy,as
explainedin sectionVI.
33
o
z
10
SNR dB
FrcunB 2.7 - NormalizedMSE of the time delay estimatefor different QAM modulation
order,usinga root-raisedcosinefilter with a roll-off factor0.5.
_
MCRLB
- < - 16-OAM withforcd symbols
- a - 16-OAMwilbullorced
1 0-
o
5
t o-'
.!
z
to'
1*.
10
SNR dB
15
FtcuRs 2.8 - Comparisonof the estimationperformancewith andwithout forcedsymbols
using 16-QAM modulationandfor a roll-off factorof 0.5.
2.8 Conclusion
A computationallyefficient techniquehasbeendevelopedto implementthe CML estimator of the time delay parameter.Basedon a$screte-time model, the transmittedsymbols
are supposedto be unknownandno restrictionon their distributionwas assumed.To avoid
iterative techniquesand their drawbacks,the importancesampling method was used to
find the ML solution.Its main advantageover the iterativeproceduresis that it doesnot
requireany initial guessof the time delay parameterand that it is far lesscomputationally
expensivewhile retaining good performances.Moreover, its convergenceto the global
maximum is guaranteed.Relativeto other proposedmethodssuchas the CML-TED, the
IS-basedestimatorexhibitsbetterperformanceat high SNR. In practice,the choiceof the
algorithm parametersp6 and p', is critical for the estimationperformanceand for a good
choice of theseparameters,a small numberof generatedrealizationscan be sufficientto
achievesatisfactoryperformanceand reducethe computationburden.
AppendixA
Proof of lim L'",rr(')-6,
p0-++oo
In the following, we prove that Lt",oo(r)definedin (2.I7) tendsto a Dirac deltafunction
centeredat the location of its global maximum as ps ) *oo. To do so, considerthe
generalcasewhere /(r) is an integrablefunction having one global maximum, denoted
a:
(2.42)
a: arg max f (r).
r€lR
And denotingby F(r) the following normalizedfunction :
exp{pe./(r)}
E ' (* \ -
(2.43)
J'rexp{pof(u)}du'
where.I is the definition domainof F.(.). Then,for a given real numberb I o, we have :
/ A \ -_
'r ,\u/
j,
exp{p6l(b)}
"pQJ6ld"
--
exp{p6l(a)}
(2.44)
I ""p{pd(-')}d'
However,since/(a) is the maximumvalueof the function/(r), then f (b) - /(o) is a
negativenumberand,therefore,exp{ps(f (b) - f (t))} tendsto 0, as well as F'(b), when
p6 tendsto oo. As a result:
(2.4s)
lim F(r) : Q,
p0--+oo
for any realr f
a.Moreover,if we considerthat
,lim
F(a) : 0, thenwhateverr € IR,
r*m
we have lim F(r) : 0 and lim I
po-+oo
po >@
J _*
sumptionthat /];
f 1u7a'u: 0, which is in conflict with the as-
F(u)d,u:\.
^+rc
Finally,we conclude
that
a(o) I 0 andrt*"
F(u)d,u: 1, F(r) be,ll*
olTL /_*
comesa Diracdeltafunctioncentered
at a whenp6tendsto +oo.
35
Appendix B
Justificationof the approximationAT A, = * t ^
The diagonalelementsof Al A" are the convolutionof the sameshifted versionof h(.)
(lAT A"l,,n : aT ?)ao(r) foli : 0, L,. . .K - 1 whereai(r) is definedin (6)).Whereas,when the shift is not the same(i.e., lAl A,]tj : al (r)a1Q), i. + j), the valueof
theconvolution,o,f,(r)ai?), is, accordingto the Nyquistcriteria,equalto zero.However,
s i n c e w e t a k e s o m e s a m p l ehs(o. )f, I A T A , ) o , i : a T ( r ) a i ( t ) , i + j i s n o t r e a l l y e q u a l t o
zerobut still very small and negligible in front of lAl A,l;,;. To clarify, we plotted in Fig.
2.9 thefollowingthreefunctions: 96(r) : h(r) x h(r) : h(r)2, 9lr) : h(r) x h(r -T)
andg2(r) : h(r) x h(r - 2T) and we takefor example[ATA"]t,, : aT?)az(r) :
DY-j9r(mT"). Then, notice from this figure that somesamplesof gr(.) are negative,
which will compensatefor positive samplesit D#:.t 7t(mT") from the samefunction
therebyresultingin very low convolution.However,the samplesof 96(r) are alwayspositive and thereis no compensationeffect in lAl A,11,, : DX_--,lgo(mT") addedto the fact
thatgo@) > 9{r), this resultsin the following conclusion: lAlA,)t1 > IATA")t,t.
Using the sameargumentsfor the other off-diagonalelementsof A! A" leadsto the following approximation:
ATA,='irn
(2.46)
,-,-.- -
o
::
-r'"'"
l
t/r
FIcURB 2.9 -Plot of 90(.),91(.)and 92(.).
36
AppendixC
Proof of the effectof p\ on g'o,ro
In the following, we briefly showhow p', (o, equivalently p1)canrendereii (.) more peaked aroundits global maximum.Startingfrom g'o,(r*), definethe function H,"(pr) :
g'ri(r.) :
H (p't) :
(r.)}
exp{p'rr,9)
f.
(2.47)
I ""p{p\Lf)(u)ldu
J.I
where r* is the true time delay value to be estimatedand ff) 1r) is the approximation
of L.(y;r) definedin the right-handside of (22), i.e., fY)(r) : yH A,Afy. The first
derivativeof H (p\) with respecttopl is givenby equation(2.48).
H'(P'r):
-exp{p'rr1)
(r")}
rP)(r.)e*p{p\r?)
@)e*p{p'rr'9)
@))au
@)}au
?.)}J
J,exp{p'rrf,)
,r?)
@}d,)
(1,"'r{o',LY)
e*p{p\L9)1"-I1 (r!") ?. )
l
au
{p'rLf,)1117
{p',Lf,)(u)}du- l r ty, rr) exp
rexp
(q}d')
(f ,"'o{o'LY)
(2.48)
And noting thatr* - arg ma*, Ll!) 1r). it follows that
[ *'@)exp{p'rLlilluyau< rf)g.1 J[L "*o{r',Lf)(u)}c]u.
(2.4e)
Jt
ThereforeH',- {pr) > 0 Vpi . Hence,H,. (pr) is an increasingfunctionwith respectto pl,
p'[') , p'lt)we haveu,-b'])) ] H,.@']')), which means7'o<n(r*)]
i.e.,for
"r"ry
g'o,.1e(r*).
We concludethat p', rendersthe objectivefunction more and more peaked
aroundits globalmaximum.
AppendixD
Methodto generateiqj!,
In this appendix,we detailhow to generaterealizationsaccordingto g'(r). First, we generateavectoru:lur,rtr2,...,ur,)ofRrealizationsuniformlydistributedin[0,1].
37
Then,we searchr.i: G'-1(u;), whereG'-t(.) is the reciprocalfunctionof the cumulative
distributionfunction(CDF) G'(r) of r definedas :
G'(")
g'r,r(u)du.
(2.s0)
Unfortunately,a closed-formexpressionof G'- 1(r) is not analyticallytractable.Moreover,
sinceG'(r) is a steep-slope
function,a fine searchto find T; as argmin,lu6 - G'(z)l is
required and makesthe processcomputationallyintensive.However, since G'(r) is an
increasingfunctionof r, the function,S(r) : lui - G'(r)l is unimodal.This observation
allowsus to adoptthe goldensearch[17] to find the locationof the minimumof S(r). The
goldensearchis appropriatefor this problembecauseit convergesafter a small numberof
iterationsand requiresonly one function evaluationper iteration.
38
Bibliographie
t l l U. Mengali and A. N. D'Andrea, SychronizcttionTechniquesfor Digital Receivers,New
York : Plenum,1997.
third edition,McGraw-Hill,New York : 2001.
l2l M.I. Skolnik,Introductionto RadarSystems,
t3l G.C. Carter,Coherenceand TimeDelay Estimation: An Applied Tutorialfor Research,Development,Testand EvaluationEngineers,IEEEPress,New York, 1993.
I4l J. W. M. Bergmans,Digital BasebandTransmissionand Recording,Boston: Kluwer Academic Publishers,1996.
andP. Lowenborg,"Time-DelayEstimationUsing Farrow-Based
t5l M. Olsson,H. Johansson,
Fractional-DelayFIR Filters : Filter Approximationvs. EstimationErrors" EUSIPCO2006.
t6l N. Noels er al., "Turbo Symchronization: An EM Algorithm Interpretation."IEEE Conf.
Commum.,Anchorage,AK, pp. 2933-2937,May 2003.
[7] S. Kay, Fundamentalsof StatisticalSignalProcessing,Estimationtheory,PrenticeHall, EnglewoodCliffs, NJ, 1993.
[8] J. Riba, J. Sala, and G. Yazquez,"Conditional Maximum Likelihood Timing Recovery :
EstimatorsandBounds,"IEEE Trans.on Sig.Process.,vot. 49,pp. 835-850,Apr. 2001.
'An Exact Maximum Likelihood NarrowbandDirection of Arrival Esti[9] S. Sahaand S. Kay,
mator,"IEEE Trans.Sig.Process.,vol. 56, pp. 1082-1092,Oct.2008.
t10] S. Sahaand S. Kay, "Mean Likelihood FrequencyEstimation,"IEEE Trans. Sig.Process.,
vol. 48, pp. 1937-1946,JuI.2000.
'A
tl1l H. Wang,and S. Kay, Maximum LikelihoodAngle-DopplerEstimatorusingImportance
Sampling,"IEEE Trans.Aerospaceand Electro.Sysr.,vol. 46, Apr.2010.
studyof conditionalandunconditionaldirection-oft\2l P. StoicaandA. Nehorai,"Performance
arrivalestimation,"IEEE Trans.Acoust.,Speech,Sig.Process.,vol. 38, pp. 1783-1795,Oct.
1990.
'A
tl3l M. Pincus, ClosedForm Solutionfor CertainProgrammingProblems,"Oper Research,
pp. 690-694,1962.
39
t14l L. Stewart,"BayesianAnalysis using Monte-CarloIntegration: a PowerfulMethodologyfor
HandlingSomeDifficult Problems,"AmericanStatistician,pp. 195-200,1983.
tl5] S. Kay, "Commentson FrequencyEstimationby Linear Prediction,"IEEE trans. on Acoustics,Speechand Sig.Process.,pp. 198-199,Apr. 1979.
t16l K. V. Mardia, Statisticsof Directional Data, New York : Academic,1972.
t17l D. J. Wilde, GloballyOptimumDesign.EnglewoodCliffs, NJ : Prentice-Hall,1964.
40
Chapitre3
Closed-FormExpressionsfor the Exact
Cramdr-RaoBoundsof Timing
RecoveryEstimatorsfrom BPSK,MSK
and Square-QAMTransmissions
rA. Masmoudi,F. Bellili, S. Affes, andA. Stephenne
for the
: "Closed-FormExpressions
ExactCram6r-RaoBoundsof Timing RecoveryEstimatorsfrom BPSK, MSK andSquareIEEE Trans.on Sign.Process.,vol. 59, no. 6, pp. 2474-2484,Jlune
QAM Transmissions",
2OTT.
DOI: 10.1109/LSP.2008.921477
IEEE (c)
Abstract
Danscet article,nousd6rivonspour la premibrefois les expressionsanalytiquesdesbornes
de Cram6r-Raopour I'estimationdu retardpour les signauxh modulationpar ddplacement
de phase,modulationi d6placementminimum et modulationd'amplitude en quadrature.
Nous supposonsque les donn6estransmisessont inconnuesau niveaudu r6cepteuret que
la fonction de miseen forme satisfaitle critdrede Nyquist.De plus, la phaseet la fr6guence
porteusessontconsid6r6es
inconnues.Le retardresteconstantsur l'intervalled'observation et le signalregu est entachdde bruit additif. Les nouvellesexpressionsmontrentque
les performancesd'estimationne d6pendentpasde la vraie valeurdu paramdtre.De plus,
ils concordentavecles r6sultatsobtenuspar calculsempiriques.
In this paper,we derivefor the first time analyticalexpressionsfor the exact Cram6r-Rao
lower bounds(CRLB) for symbol timing recoveryof binary phaseshift keying (BPSK),
minimum shift keying (MSK) and squareQAM-modulatedsignals.It is assumedthat the
transmitteddata are completelyunknown at the receiverand that the shapingpulse verifies the first Nyquist criterion. Moreover the carrier phaseand frequencyare considered
as unknown nuisanceparameters.The time delay remainsconstantover the observation
interval and the receivedsignal is comrptedby additive white Gaussiannoise (AWGN).
Our new expressionsprovethat the achievableperformanceholds irrespectivelyof the true
time delayvalue.Moreover,they corroboratepreviousattemptsto empirically computethe
consideredboundstherebyenablingtheir immediateevaluation.
42
3.1 Introduction
In moderncommunication
systems,thereceivedsignalis usuallysampledonceper-symbol
intervalto recoverthe transmittedinformation.But the unknowntime delay,introducedby
the channel,must be estimateda priori in order to samplethe signal at the accuratesampling times.In this context,many time delay estimatorshavebeendevelopedto meetthis
requirement.Theseestimatorscan be mainly categorizedinto two major categories: dataaided(DA) andnon-data-aided(NDA) estimators.In DA estimation,a pri,ori known symbols aretransmittedto assistthe estimationprocess,althoughthe transmissionof a known
sequencehas the drawbackof limiting the whole throughputof the system.Whereas,in
the NDA mode,the requiredparameteris blindly estimatedassumingthe transmittedsymbols to be completelyunknown.In both cases,the performanceof an estimatoraffectsthe
performanceof the entire system.In the caseof an unbiasedestimation,the varianceof
the timing error is usually usedto evaluatethe estimationaccuracy.The CRLB is a lower
bound on the varianceof any unbiasedestimatorand is often used as a benchmarkfor
the performanceevaluationof actualestimatorsll,2l. The computationof this boundhas
been previously tackled by many authors,under different simplifying assumptions.For
instance,assumingthe transmitteddatato be perfectly known and one can derivethe DA
CRLB. The modi{iedCRLB (MCRLB), which is alsoeasyto derive,hasbeenintroduced
in 13, 41,but unfortunatelyit departsdramaticallyfrom the exact (stochastic)CRLB, especiallyat low signal-to-noise
ratios(SNR).
Actually, the time delay stochasticCRLBs of higher-ordermodulationswere empirically
computedin previousworks.Their analyticalexpressions
were tackledonly for specific
SNR regions,i.e., very low or very high-SNR valuesand the derivedboundsare referred
to asACRLBs (asymptoticCRLBs). In fact, in [5] the stochasticCRLB wastackledunder
the low-SNRassumptionand an analyticalexpressionof the considered
bound(ACRLB)
was derivedfor arbitraryPSK, QAM and PAM constellations.In this SNR region,the authors of [5] approximatedthe likelihood function by a truncatedTaylor seriesexpansion
to obtain a relatively simple ACRLB expression.An analyticalexpressionwas also introducedin [6] underthe high-SNRassumption.
This high-SNRACRLB coincideswith the
stochasticCRLB in this SNR regionbut unfortunatelyit cannotbe usedevenfor moderate
(practical)SNR values.Another approachwas later proposedin [7] and [8] to compute
the NDA deterministic(or conditional)CRLBs, in which the symbolsare consideredas
deterministicunknownparameters.Then the conditionalCRLB is derivedfrom the compressedlikelihoodfunctionf (U;0,0) in which g standsfor the observedvector,0 is the
parametervector of interest(including the unknown time delay) and fr is the maximum
likelihood estimateof the transmittedsymbols r. However,it is widely known that the
conditional CRLB does not provide the actual performancelimit (unconditionalor stochasticCRLBs). In an otherworks, the stochasticCRLB was empiricallycomputed[9]
43
assumingperfectphaseand frequencysynchronizationand a time-limited shapingpulse.
Later in [10], its computationwas tackled in the presenceof unknown carrier phaseand
frequencyandpulsesthatareunlimitedin time.Both [9] and [10] simplifiedtheexpression
of the boundsbut ultimately resortedto empirical methodsto evaluatethe exact CRLB,
withoutprovidingany closed-formexpressions.
Motivated by thesefacts, in this work, we derive for the first time analyticalexpressions
for the stochasticCRLBs of symboltiming recoveryfrom BPSK, MSK and squareQAMmodulatedsignals.We considerthe generalscenarioas in [10] in which the carrierphase
and frequencyoffsetsare completelyunknown at the receiver,and we show that this assumptiondoesnot actually affectthe performanceof a time delayestimatorfrom perfectly
frequency-andphase-synchronized
receivedsamples.The derivationsassumean AWGNcorruptedreceivedsignal and a shapingpulse that verifies the first Nyquist criterion. The
last assumptionis verifiedin practicefor mostof the shapingpulses.
This paperis organizedas follows. In sectionII, we introducethe systemmodel that will
be usedthroughoutthis article. In sectionIII, we derive the analyticalexpressionof the
stochasticCRLB for any squareQAM modulation.Then,in sectionIV, we outline the derivationstepsof the CRLB in the casesof BPSK andMSK transmissions.
Somegraphical
representations
arepresentedin sectionV and,finally, someconcludingremarksaredrawn
out in sectionVI.
3.2 SystemModel
Considera traditional communicationsystemwhere the channeldelays the transmitted
signal and a zero-meanproper2 AWGN, with an overall power o2, cor:rttptsthe received
signal.In the caseof imperfect frequencyand phasesynchronization,the receivedsignal
is expressed
as :
,y(t):
+,w(t),
J n" "(t - r)ejQ"f"t+o)
(3.1)
where r is the time delay, I is the channeldistortion phase,/" is the carrier frequency
offset and j is the complex number verifying j2 : -I. The parametersr, g andf" are
assumedto be deterministicbut unknown.They canbe gatheredin the following unknown
parametervector :
':l',0,f.)'
(3.2)
In (3.1),'u(t) is a proper complexGaussianwhite noisewith independent
real andimaginaryparts,eachof varianceo2f 2, andr(t) is the transmittedsignalgivenby :
2Apropercomplexrandomprocess
n{u(t)2} :0.
o(l) satisfies
44
K
\-
r ( 1 ): L o , h ( t - i T ) ,
(3.3)
i:I
with {a,1}[, beingthe sequence
of K transmittedsymbolsdrawnfrom a BPSK, an MSK
or any square-QAMconstellationand 7 is the symbol duration.The transmittedsymbols
are assumedto be statisticallyindependentand equally likely, with normalizedenergy,
i.e., E{lall'} : 1. Finally, h(t) is a square-rootNyquist shapingpulse function with
unit-energywhich will be seenin sectionsIII and IY as would be expected,to have an
importantimpact on the CRLB and thereforeon the system'sperformance.The Nyquist
pulse9(l) obtainedfrom h(l) is definedas :
s(t):
/n F-
J__
h(r)h(t-rr)dr.
(3.4)
and satisfiesthe first Nyquistcriterion:
(3.s)
g ( n T ): 0 , n 1 0 .
O,of thevectorufromthereSupposethatweareabletoproduceunbiasedestimates,
ceivedsignal.Thenthe CRLB, which verifiestr{(t > CRLB(z), is definedas [1,
")"}
2l:
CRLB(z) : I-r(u),
(3.6)
where1(z) is the Fisherinformationmatrix (FIM) whoseentriesare definedas :
:E
u,t : r,2,r,
tr(,)tn,i
{W W},
(3.7)
with L(u) beingthe log-likelihoodfunctionof theparameters
to be estimatedand {"r}?_-.
are the elementsof the unknownparametervectoru.
To beginwith, we showin AppendixA that the problemof time delayestimationis disjoint
from the problemof carrierphaseandfrequencyestimation.Indeed,we showthat the FIM
is block-diasonalstructuredasfollows :
/ cnl-n-'(r; o
I("): I
\
0
h(0.f.)
( 3.8)
)
whereO : [0 , 0]", CRLB(r) is the CRLB of the time delayparameterandI2(0, /") is the
(2 x 2) FIM pertainingto the joint estimationof /. and 0. Hence,we prove analytically
that we dealwith two separable
estimationproblems;on onehandtime delayestimation
and on the other hand carrier phaseand frequencyestimation.Actually, this conclusion
has been alreadymadein [10] but the authorsresortedto empiricalevaluationsto find
45
that the elementslI(r)|t," and [,t(z)]1,sof the FIM are almostequalto zero.Now, since
the parametersare decoupled,we only needto derivethe lirst elementof the global FIM,
V @)lt,t in order to find the CRLB for time delay estimationunder imperfect frequency
andphasesynchronization.Therefore,in the following, we considerthe virtually derotated
receivedsignal/(f) givenby :
i@
:
Y(t) s-lQ*l't+e1
:
J F+r(t - r) + fr(t),
(3.e)
where6(t) : u(t)e-i{z"t"t+d) i, alsoa proper AWGN with an overallpowero2 sincethe
nuisanceparametersare assumedto be deterministic.
We mentionthat l. l, n{.}, S{.} and {. }- returnthe magnitude,real,imaginaryandconjugate of any complexnumberand E{.} is the statisticalexpectation.We also definethe
SNR of the systemasp - E"lo' .
3.3 Time Delay CRLB for Square QAM-Modulated Signals
In this section,we introducethe main contributionembodiedby this paperwhich consists
in deriving closed-formexpressionsfor the stochasticCRLBs of time delay estimation
whenthe transmitteddataareunknownanddrawnfrom anyNI -arysquareQAM-constellation
(i.e.,M :22P).
Before further development,it is important to emphasizethat an exact representationof
vectorrepresentation
/(t) requiresan infinite-dimensional
/. But let us considerthe -A/dimensionaltruncatedvectorsi x , n N and 61,', representingthe projection,over an orthonormalbasisof N dimensions,of /(t), r(t) andfr(t), respectively.
Then,the pdf of iw
conditionedon the transmittedsymbolsa and parameterized
by r is given by [4] :
/Y t
^.-^|
:!+ft"v\P(ilNla;')
la*-t*l'\
"-: I
( 3.10)
To derivethe likelihood function which incorporatesall the informationcontainedinfi(t),
problemsapwe shouldmakeN tendto infinity to get P(ilo;r). However,convergence
-'".l
pear.To overcome
r) is dividedby Lf Qro2)tu*p
theseproblems,
P(fr1,,1a;
''k:rlA*l-J
^ {t t I o' D*'
to obtain :
/_N
: exp
^@lo;r)
1'+I
a"
'
Tb'*r')
n{g*,;}
(3.11)
k:L
I
and as .A/tends to infinity, we obtain the conditional likeli hood function :
^@to;r):
exp
{+
f_l
ntaft)r(t).dt}
3
46
(3.r2)
I-:@(')t"dtj
To begin with, we note that sincethe transmittedsymbols{oo}f:, are equally likely, then
the desiredlikelihood function of the derotatedobservationvectorfi canbe written as :
: "{frr'(,,,0(,))}
^(y;,)
,
( 3.13)
wherethe expectationis performedwith respectto the vectorof transmittedsymbolsand
F(ao,y(t))exp
{+/_-
- ir - r)d,t- tirr'}
*rn,t)ai}h(t
el4)
It canbe shownthat (3.13)reducessimplyto :
1K
^(i;"):#fla'(').
(3.1s)
;-1
where
(
E"t
2JT
,,, \
- = l r ^ 1r ", )+ - - #
H
; ( r ) : \Lo'
" * p i( o cpcc
r'*-
/
J-a
)
n { i ( t ) 4 } h ( t- i T - r ) d t l , ( 3 . 1 6 )
)
in which C is the constellationalphabet.Actually, the main difficulty in deriving an analytical expressionfor the stochasticCRLB stemsfrom the complexity of the log-likelihood
function.Therefore,we will manipulatethe summationinvolvedin (3.16).In fact,considering only squareQAM-modulatedsignals,we are able,by exploiting the full symmetry
of the constellation,to factortzeH;(r) which in turn linearizesthe globallog-likelihood
function and ultimately linearizesall the derivations.
Indeed,denotingby C the subsetof the alphabetpointswith positivereal and imaginary
parts(i.e.,e : 1pl - 7)do+ jQk - l)drlo,r:.r.,2,...,2p
r), the constellation
alphabetC is
decomposed
asfollows :
c:e u i. u (-i) u (-i.)
(3.r7)
Note that do is the inter-symboldistancederivedunder the assumptionof a normalizedenergysquareQAM constellationas follows :
(3.18)
2nli"_i Qk -
Using(3.17),we rewrite(3.16)as :
H,(r):t "*o{ _.u:rca'},
("*o{ry,,f
\
Zt"ee
- ,, n{r(/)(-d)}h(r
"tor}
+*
( 2{E; f
-_ i r _ , - ) d 1 }
*exp
\TJ_*m{t(/)(-ek)}h(L
)
2\/4
^_-^|
[+*
(t - ir , _ ,)dr]
*, exp
)e;}h
J-*n{t(f
\T
)
(2J4[t*
..)\
(t - ir - ,)dt
* exp
J_*n{t(r)Ck}h
\T
})
Now using the hyperboliccosinefunction definedby 2 cosh("r) :
(3.re)
e' + e-', (3.I9)
reducessimply to :
H?):2L",.0 -*vrf}I,*^(+l_*-*1y1,;u;
}h(t- ir - r)d,t\
o" J-o'
oz'
{
t
)
L
\
/
?a€c
*cosh
(#1.-"{aft)ek}h,(t- ir -,)")]
(3.20)
Moreover,usingthe fact that cosh(o)f cosh(b): 2 cosh(#) cosh(f ) andnotingthat
En * eI : 2ft{Zx} and1p - 6T": 2j3{ck}, weobtain:
- ,, - io,) *
Hiu):2D.*o{ -*lu-f},",n(+in{er}/ ?tn,,)}h(/
ci'eu
- u,- oot)
"",n(+s{er}/_1r0,,,}h(t
(3.2r)
Recallthate : {Ql-l)do+ jQm-I)do},,^:r,2,...,2e-tandhencethepreviousexpression
of HJr) is rewrittenas :
E"((21- t)' + (2^ - r)') d7
^tP-'"ot
2P-L 2P t
Ht(") :
(
)"
"'"n(+
- ir - ,lot)
et- r)o,I_J*ro@)h(t
"
"""n(+
- ir - iot)
e* -1)d,/% {i(t)}h(t
(3.22)
Then,splittingthe two sumsin (3.22),Ho(t) is factorizedasfollows3:
3Notethat similar factorizationwas recently usedto derive an analytical expressionfor the NDA SNR
estimation
t111,t121.
H n(r): 4 tr (u lr)) .P(%(r)),
where
r*-
tJ;(r):
v,(i :
/_."
r*-
J_*
(3.23)
m{utt)} h(t - iT - r)dL.
(3.24)
s{t(r)}h(t- ir - r)dt,
(3.2s)
and
2P-1
:
,F-l
. \ rr\) _
/
-
'
-*en\- exp
^ {
Z_
02'
rlra3}
u ) "orr,l'
\
t
k:r
/^
\
t=
'$ek
02
- r)d"r\
.
'"
(3.26)
)
Now, injecting the expressionof Hift) in the likelihood function of the receivedsignal
(3.15),we obtain:
t
,r \K
K
^@;,): (+ ) il F(u/r))F(V(i)
\wt /
e.27)
i:,
Finally, the log-likelihood function of the receivedsignalexpandsto :
L(r):Dr"(r(u,("))) +th(F(%(?)))
i:l
Q.28)
i:l
Note from (3.28)that dueto the factorizationofHift) in (3.19),the globallog-likelihood
functionof interestin (3.28)involvesthe sumof two analogous
terms.This reducesconsiderablythe complexity of the stochasticCRLB derivation.
In fact, the first derivativeof (3.28) with respectto r is obtainedas follows :
ygt.ryP. i\y:g!!W)
4r:f
dr
rluntD
0r
P(V?D Er
2
(3zs)
%P i, givenby:
wherei(r):
qD-L
F(r):
- rya?,}'#er- Ddo,t"n(Tek_1)d",)(3.30)
D ""0{-p(zk
I^-1
Then the first diagonalelementof the FIM matrix is expressedas :
tfqo)']:u{f)a{q,?
!u''tt
: EltLLriffiffiui(r)u1(r)l
Ir(")h:,:El (#
\ or /) |
1
)
)+
P-l:F@mrui#ut(r)ut(r)]+
,,{fi #ffiffi uG)v(r)
.
} "{*t #&ffi tn)i,,(.)\
(i:tt:t
'//'\'L\'))
J
ti:It:l
' \'r\'
( 3 . 3l )
49
whereU/r) andVl(r) arethe derivativesof LI1(r) andI{(r) with respectto r.
Startingfrom (3.31),the derivationof [/(")]t,t involvesthe evaluationof threeexpectations.However,it is easyto verify that the first andthe last expectationsin the right-hand
sideof (3.31)areperformedwith respectto two randomprocesses
havingthe samestatistical propertiesand they are thereforeidentically equal.Moreover,as shownin Appendix
B, the secondexpectationis equalto zero.Therefore,(3.31)reducessimplyto :
: 2ii'{ " :99}ggR
r,1,;r,i,)
rr(,)t,,,
},
F
(
U
A
,
r
(
U
r
(
r
)
)
n
i=r r:r I
)
(3.32)
First,we considerthe casewherei : l, andweshowin AppendixC that Ui(r) andUi(r)
are statisticallyindependent.This resultsin :
( r\r,J),),)',*,,,,,]
: u { f y'@,,'J'}
(333)
u
[
u{ (a,r,r),}
"\\4urlll\''\//J-\\F@Gt/lr\
/)
Thesetwo expectations
involvedin the right-handside of (3.33) are easilyevaluatedas
follows :
r*m
, | (F(u1(r)))')
"t(iffi,)]: J-I
J;P
-lutot
: +ii'((i,-k)r)
{(ar"l)'}
"
(3.3s)
k:1
where 9(.) and g(.) are the first and second derivative of g( ), respectively.We simplify
(3.34) by changine t/rUo(t) lo by r and we obtain the following result :
(3.36)
where
2p- r
{-p(zk- 1)'d3}Jrek - l)d,sinh( /zp1zr,- r)d,r),(3.37)
"*o
k:1'
sp(r): I
2pt
Gr\t) :
\-a
(
Lexp {
-p(2k - t)'a?}
k:L
"otn
(r/-zoQk- t)d,or).
(3.38)
50
We now considerthe casewherei, I l.The intersymbolinterference
resultsin a statistical
dependence
betweenUi(r) andthelirst derivatives(Ii(r) andt{(r) (ikewise for V1(r) and
anaUO>. Thus using a standardprobobility approachto derive
the expectations
involvedin (3.32),we first average
by conditioningon U;(r) andUt(r),
the first derivativesl4(l
then averagethe resultingexpressionwith respectto thesetwo random variables.To that
end,considerthe expectationof U;(r) and U7(r) conditionedon U1(r) andU{r) :
tr{ui(r)lUn(r),u{r)} : U/r) g((i - t)T),
(3.3e)
E {U1(r)lU
u(r),U,(r)}: Uo(r)g((t- iln.
(3.40)
Using (3.39)and(3.40),it follows that :
,
t" { Fi\!,fllgg9
rr,G)utu)lro?),1,,(,-)}
ilyl!'ll
ily:!'ll
/ 1 'I \ ' / : F
'r
( u ; ( r )F
) (uilr))
( u 1 ( r )")
( u t ? ) )F
I
J
- t)r)s((t-d\T\)
ua(r)u{r)s((i,
and we obtain :
t.l
) _ ( ,tr,t,ll,r,(,)))'
rp,1,)r1y,1-rr
: ,!
u,ft)rrtft)]
n,((,_t)r1,.4z)
t"iGtFdffi
\- [ora(,))-,) /
wherethe last equalityfollows from the statisticalindependence
of Ui(r) and t{(r) and :
# I-.J.n^.7"-*a.
"{ffiffi,?)}:rl
(3.43)
Finally, gatheringall theseresults,we obtain the analyticalexpressionof the stochastic
CRLB for symboltiming estimation.From squareQAM-modulatedsignalsin thepresence
of carrierphaseand frequencyoffsetsas follows :
:
g'(-- n)r)-2Kps(o))
cRLB(r)
tt
i
'
l(rnf
m:rn:r
L\
/
""2f #"-*d,
2 K K
l-1
-,)r)l
(3.44)
*s,1'7"-*a*)
(l-:
tt i'(^
#
m:ln:I
I
Note that for large valuesof K, one can usethe following accurateapproximation[10] :
KK+m
tI
g"(^-n)r) x K \, t'@r\.
(3.4s)
It is worth mentioningthat the new analyticalexpressionin (3.44) allows the immediate evaluationof time delay stochasticCRLBs, contrarily to the empirical approaches
presented
in [9] and[10], andthis is madepossiblefor anysquareQAM modulationorder.
Second,the shapingpulse is involvedonly via 1j(0) and S'((^,- r,)T), and is separate
from the factorsresultingfrom the modulationorder.Moreover,to the best of our knowledge,we showherefor the first time, throughour new analyticalexpression,that the true
value of the time delayparameterdoesnot affect the actualachievableperformanceas intuitively expected,i.e., the varianceof the estimationerror holds irrespectivelyof the time
delay valueto be estimated.
3.4 CRLB for BPSK and MSK Modulated Signals
In this section,we considerthe BPSK and MSK modulations.In BPSK transmissions,
thedatasymbolstakevaluesin {-1, +1} with equalprobabilities.In MSK transmissions,
the symbolsare definedas a7"11: j ancnwherecp is a sequenceof BPSK symbolsand
a6 is the originalvaluedrawnfrom the set{-1, -i,+7,*7}. For thesetwo transmission
schemes,the key derivationstepsof the NDA CRLB will be briefly outlined in the following. All derivationdetailscan be found in Appendix D.
First, the likelihood function of interestbasedon the receivedsignalis :
K\
. ) d ,+t * l u , l ' ) ,
j:l
\
o'
J-x,
G.46)
/
wherebi is equalto 1 and jo-'oo for BPSK and MSK, respectively.
Therefore,we show
that the useful log-likelihood function of fr is given by :
L(r):ir("",n
(+ I_:n { b : i ( t ) } h ( t - f f - , ) " ) )
G.47)
Note that /(t) is definedin (3.9).After somealgebraicmanipulations,
detailedin Appendix D, it turns out that the analytical expressionof the stochasticCRLB for time delay
estimationis the samefor BPSK andMSK modulations,andit is givenby :
CRLB :
- ftu,pe))
('f, iuu^-n):r)f,r'r)
[^,{('
,f,ig((m-',t,)]
tn:l
n:l
(3.48)
where0(.) is definedas :
,2
: l"* cosn, / -2P t \ * - '
oQ)
\t/
)
,-U
52
,n
(3.4e)
3.5 Graphical Representations
In this sectionowe provide graphicalrepresentationsof the time delay CRLBs and the
CRLB/MCRLB ratio for different modulationorders.-First,we mention that the evenin^2t,t
,2
ro
,2
tegrandfunctions
involvedin (3.44)and (3.49).
rgr(r)e-T and -;ip-1
ffi"-+,
respectively,decreaserapidly as lll increases.Therefore,the integralsover [-oo, foo]
can be accuratelyapproximatedby a finite integral over an interval [-A, A] and the Riemannintegrationmethodcanbe adequatelyused.In our simulations,we notethat A : 100
and a summationstepof 0.5 providedaccuratevaluesfor the infinite integral.
0
5
sNR [dBl
l0
FIcunB 3.1 - Compressionbetweenthe empiricalCRLB and the analyticalexpression
pulsewith
in (3.44)for differentmodulationordersusing K : 100 and a raised-cosine
rolloff factorof 0.2.
First, we plot in Fig. 3.1 the CRLBs for different modulationordersandcomparethem
to the onespreviouslyobtainedempiricallyin [10]. We seea good agreementbetween
the two approachestherebyvalidatingthe developmentsabove.Then,we confirm through
Fig. 3.2 that, at low SNR values,the MCRLB is a looserbound comparedto the exact
CRLB. Indeed,this figuredepictsthe CRLB/MCRLB ratio asa functionof the SNR.This
ratio quantifiesthe performancedegradationthat arisesfrom randomizingthe transmitted
dataandit approaches
1 at high SNR values.Hence,in this SNR region,the MCRLB can
be used as a benchmarkto evaluatethe performanceof unbiasedtime delay estimators
insteadof the exact CRLB, since it is easierto evaluate.However,the gap betweenthe
two boundsbecomesimportantas soonas the SNR dropsbelow 7 dB, evenfor QPSKmodulatedsignals,where the stochasticCRLB quantifiesthe actual performancelimit.
53
Moreover,we considerin this figure two valuesof the rolloff factor, 0.2 and 1, in orderto
illustratethe effect of the rolloff factor on timing estimation.Clearly,timing estimationis
lessaccurateat a lower rolloff factor (largerintersymbolinterference).
Moreover,we seefrom Fig. 3.3 that the different CRLBs tend to ultimately coincide
with the MCRLB as long asthe SNR getsincreases.Actually, in the high SNR region,the
achievableperformanceof NDA estimationof the signal time delay is equivalentto the
one obtainedwhen the receivedsymbolsare perfectly known sincein this SNR rangethe
MCRLB coincideswith the DA CRLB.
J
o
=
J
o
FtcuRe 3.2 - CRLBA4CRLBratio vs. SNR for differentmodulationordersusins K :
100anda raised-cosine
pulsewith rolloff factorof 0.2 and 1.
In the speciliccasewhereh(l) is time limited to the symbolduration,the corresponding
CRLB follows directly from the generalexpressionin (3.44) by taking it(mT) : 0 for all
mtL:
1 -r
cRLB(rpl-r* oU1o1
_^T2 1
drl
(3.50)
I
Note from (3.50)that the resultingCRLB becomesthe productof two separateterms;
one dependingon the shapingpulsefunction and the other on the signalmodulation.This
specialboundis plottedin Fig. 3.4. We seeagaina good agreementin this specialcase
betweenthe CRLBs obtainedfrom our analyticalexpressionin (3.50) and their empirical
plottedin Fig. I of [9]. This particularexpressionstill finds applicationsin
counterparts
54
5
sNR [dBl
FIcuRe 3.3- CRLB vs. SNR for differentmodulationordersusinsK : 100anda raisedcosinepulsewith rolloff factorof 0.2.
manyconventionalsystemsandin the emergingimpulseradiotechnologylI3,l4l where,
precisely,synchronization
standstodayas a very challengingissue.
Fl
o
j
Q
sNR IdBl
FtcuRs 3.4 - CRLB/IvICRLBratio vs. SNR for differentmodulationsand a time-limited
shapingpulse.
55
3.6 Conclusion
In this paper,we derived,for the first time, analyticalexpressionsof the Cram6r-Raolower
boundfor symboltiming estimationin the casesof BPSK,MSK andsquare-QAMmodulations.We consideredthe stochasticCRLB wherethe transmitteddata are unknown and
randomlydrawn.The carrierphaseandfrequencyoffsetsarealsosupposedto be unknown
(nuisanceparameters).We showedthat the knowledgeof the phaseandfrequencydoesnot
bring any additionalinformation to the time delay estimationproblem and that the latter
is decoupledfrom thejoint estimationof the carrier frequencyand phaseoffsets.Moreover, our analyticalexpressionsfor the CRLBs underlinethe fact that theseboundsdo not
dependon the time delay value,which usedto be statedonly intuitively.We confirmed
also that the modified CRLB is a valid approximationof the exactCRLB in the high SNR
region and that it can be usedas a benchmarksinceit is easierto evaluate.Furthermore,
the derivedanalyticalexpressionscorroboratepreviousworks that empirically computed
the stochasticCRLBs via Monte Carlo simulations,and henceprovidea usefultool for a
quick andeasyevaluationof the CRLBswith BPSK,MSK andsquare-QAMmodulations.
Appendix A
Proof of the Block-DiagonalStructure of the FIM
To show that r and u :
lf ",01r aredecoupled,we considerthe actualreceivedsignal
y(l) insteadof the virtually derotatedsignal9(l). Then we follow the samederivation
stepsfrom (3.13)to (3.28)to retrievethe log-likelihoodfunctionparameterized
by z as
follows :
KK
L(u): It" (r1uo1u7\
+ t rn(F(V(u)))
i-l
(3.sr)
;-l
The first derivativesof this functionwith respectto the Ith elementof u, {u1}!}1, andr
are,respectively,given by :
aL(u)
_:
0'tr1
K
F (Ut( u) )\ur ( u) - p( U( ") ) aV@) t _ 1 e
1 u - L' '1
\./--) F ( U a ( u ) )
;-1
0tt1
P(V@))
7ut
(3.s2)
and
|L(u) :_ aa pv,@)) \tLi@)_
a,
ev@)))Vi@)
- e(v@))- #,O T
FW,@D
a,
k
where F.(.) is defined in (3.30). Then we average
followine result :
56
0L(v) }L(u)
0r
AUQ)
( 3.53)
as in (3.31)to obtainthe
:'
[1(rz)]r,r+r
{%?u#P}
:'ii'{ffi8#&ry#.#}
i:l
m:I
['
G54t
'
In order to simplify the calculations,without loss of generality,we considerI :
1. To
beginwith, we lirst differentiateU^(u) with respecttof . andwe obtain:
)U*(u) _r- [**
-""
3 {s(r)"-it2nr"t+o}
hft - mT - r)tdt
af.
J-*
K
-- . )'"- \ - c > /m - 1
{o*}l** n(t- nr - lh(t - mT- irar+ l**s{w(t)}h(t- ntr - r)tdt.
(3.55)
is a function of the imaginary part of the transmittedsymbols and the derotated
W
noise,which are mutually independentfrom the real part of the transmittedsymbolsand
the derotatednoise.As a result,
is independent
from Uo(r),U*(r) anO$p.
W
allowsus to split the expectations
in (3.54):
tfrt
0u^(u)l : - | rg,@))p(u^(v))
au;@)]
- [ rg,p\ rp*p)) lui@)--d;
'\NpfiTW;6
"
F(IA"D a,
I F1rl,(t,D
a, I
J
u {0,:(")\
Ldur )
(3.s6)
Notingthatthelastexpectation
is equalto zero,itfollows immediatelythatll (u)]r;zis also
equalto zero.Thus, we show analyticallythat the two parametersr and f . aredecoupled.
The samemanipulationsare usedto prove that r and0 arealso decoupled.Therefore,the
FIM is block-diagonalstructuredasgivenby (3.8).
Appendix B
-- o
l\Yt?\fr,(r\v(r\
Proofof
vL u
tr {| . r+\y=,v)!
\
)' L\' ) }
trrrt")) F(v1i))'r2\'
r rvvr
J
rn thefoilowing,
webrieflyshowrharE
{ffiEwe
u,?)utr)} : 0.By definirion,
Ui(r) dependson the real part of /(t), while I{(r) involvesthe imaginarypart of T(t),
which are statisticallyindependent.
It follows that Ui(r) and\fi(r) are independent.
The
(Ii(r)
sameargumentshold to show the statisticalindependence
andV(r) Then, it
of
immediatelyfollows that :
),
E
u{i,\y;9,),{(r4('\t
b {!!yI{/,(,)}r{::y:fllai,y}rr,
". ,,J
tFdiqUDuift)v1i) j: tF(ui;Du,\')l-lF(v1(r))
57
(seeAppendixC), eachwith mean
And sinceU6(r) and(I;(r) arc statisticallyindependent
zero.we obtain:
:o
"{ffi8ffi,,?)v(,)}
(3.s8)
AppendixC
C.l - Pdfsof Ui(r) and V?)
In this Appendix, we establishthe joint pdf of U6(r) and Vi(r) defined,respectively,in
(3.24) and (3.25).To that end, we deline theproper complexrandomvariableZi(r) :
- r)dt.Itcanbe easilyseenthatzi(r) : U6(r)+ jvo(t) andthat
[i|y,Al hQ iT
P(Z/r)) : P(Ui(r),V(r)) Usingthesamealgebraic
manipulations
from (3.16)through
(3.23),we establishthe pdf of Z;(r) as follows :
P(Z;(r)):
:
1 1 ,xp{-V!4!}n,(,)
M nor,
#.*o
o.
I
{
)
-u?(r)j-v'2(r)}
o,u,',,F(%('))
: P ({Ji( r ) P(
) V( ") ) ,
( 3 .s e)
where
P(ui(r)) :
(3.60)
p(v(,)):#l#"r{ YP},,nr,
(361)
Note that the factorizationof the joint pdt P(Ui(r),V(r)) of Ui(r) and Vi(r) to their
elementarypdfs confirmsthat thesearetwo independentrandomvariables.
C.2 - Proof of StatisticalIndependenceof Ui(r) and Ut(r)
First,note that,Uij) canbe writtenas :
(Ii(r):
where
tf4*n{a} + l3t,
(3.62)
n{ ?r ( r ) } h(-, iT - r ) dt.
(3.63)
u,:I_.:
58
Therefore,Ua(r) is givenby :
*
r+oo
t
LIi?) :
m{o-}s((i-m)T)- /
f
/
--r
n 1 a 1 r t } l , t r -i T - r ) d t
-OO
K
: D n{o-iift,- ^)r) - 00.
Q.64)
m,:I
In addition, ffi{rr} andp6 are independentsince the noise and the transmittedsymbols
are independent.
Recallalsothat 9(0) : 0 (the maximumof g(r) is locatedat 0). Then,
- m)f) and ft{r4} are alsoindependent.
Moreover,[)6 andpi are obD[:rn{a*}g((i,
tainedby a linear transformationof the Gaussianprocessn{tu(t)}. Hencethey are also
Gaussianprocesses.
Then, sincethe cross-correlation
of Bi andp; is equal to zero, as
shownbelow :
r
'-tnA\
LIu,uil - tr]
\
(i*r)?*r
f f
I I
t/
iTIr
f t { u ' ( f 1 ) e - i Q " I " t t + d ) 1 n 1 ' , ' i t r ; " - ) ( 2t d
t f)' }t zx
)
,:,h ( 1 ,- i T - r ) h ( 1 2- i T - r ) d L f i L 2
j
62
f f l*
: + II
z
d(t, - t2)h(t1)h(t2)dtdt2
J J-x
fi2
: -;f0 )
-n
(3.6s)
andtherefore
then,pi andUi(r) areactuallytwo uncorrelated
Gaussianrandomprocesses
theyareindependent.
Thus,Ui(r) andu6(r) areindependent.
Appendix D
Derivation of the Analytical Expressionsfor the CRLBs in
Caseof BPSK and MSK Modulations
Startingfrom the expressionof the log-likelihoodfunctiongivenin (3.47),we will consisignals,
der the two casesof BPSK and MSK separately.
Startingwith BPSK-modulated
we showthat the log-likelihoodfunctionin(3.47) reducesto :
:f ,"(*-n(Tr^,,)),
L(r)
(3.66)
where
u6(r):/J
*,n,t)j h(t- ir - r)dt.
59
(3.67)
Then, the first derivativeof the log-likelihood function with respectto the time delay parameter,r, is given by :
" 'i"n (@",u,(')) uili.
a, o, +;i@id"
oL(r) -2/4
(3.68)
wheret/,(r) denotesthe first derivativeof tl(r) with respectto r. It is easyto seethat :
(3.6e)
Now injecting(3.68)tn(3.7), we obtain:
17 K
K
frl,,,: 4+tf
ot?u*-
:
(
/q
trF-
\
/o.ff
.
.\
,l
e {tu"n1"$uft)" / )tanh("f#u'tr))unG)u,(rt>
'")
"
/
0
2
o
'
\
\
[
KK
(3.70)
+>)Duo,,.
04L
t:l
l:l
Notethat (3.70)is similarto (3.32)(obtainedin thecaseof squareQAM modulations).
Thus,for the samereasons,
it is moreconvenientto separate
thecaseswheni : I andi I l.
Moreover,it can be shownthat the pdf of Ui() is given by :
:
P(u6(r))
.o,r'
(#run)
# ", {-qg*}
( 3 .7t)
Thus, it can be shownthat, after somemanipulations,the expectationsinvolved in (3.70)
reduceto :
{-?} ['* "^n'
\ : "*p
n
"t""""
{ran.,'fltr\
lTQe"p
{-4\
"^"\
o'I*o,
o'-)J
,/"r, J---""rt\ry")
\
:
(3.72)
1 - ' - ' 9 1 0 1" ,'
t/2r
KK
: E"t t g'(^ - n):r)E {(uc(r))'}
m:I
,
o-
' 2 q" "( '0, ) .
(3.73)
n:L
-i,(i- t)r)(# I_:r,1"h
(ry) "*o
{ S}",)'
-ps'((i t)r)
Q.74)
Finally,we obtaintheclosed-formexpression
for thestochastic
CRLB of BPSK-modulated
signalsas follows :
: lnrl(, - $oo))
(, i i s,((^- n)r)-frrol)
CRLBsp5K
"
t
/
2
t
r
/
/
L'L\
\ m:tn:l
K
ptI
K
ll-1
g(,,-")r)ll
m:7n:l
(3.7s)
JJ
where0b) is definedin (3.49).
Now, consideran MSK-modulatedsignal.In orderto find the derivativeof (3.47)with
respectto the time delay r, we needto separatethe caseswhereb; is real or imaginary.To
do so, we assume,without loss of generality,that K is an evennumber(r.e.,K -- 2P) and
ao :7. Using theseassumptions,
the log-likelihoodfunctioncanbe writtenas :
P
/
/q.rp
\\
L(r): I,n (cosh
lTr,,-,(,r))+
/
/2{8,,,.\\
h (cosh
, e.76)
lffu,,tn ) )
where
uo(r):
/o+-
n { t ( r ) }h ( t - i T - r ) d t ,
J_*,
(3.77)
and
v(,) : /*- *{i(, )} h(t- ir - r)d,t.
(3.78)
J-a
Then,the first derivativeof (3.76)with respectto r is givenby :
ry :'+t
)*ranh
(+r,,-,G))uzi-t
(*u,,,t) u,o)tr,
( r^'n
with U26-{r) and ho(r) being the derivativesof (}2i-y(r) andV26(r)with respectto r,
as :
respectively.
Then,the first diagonalelementof the FIM matrix is expressed
[/(,)],,,:u{
(ry)'}
+*te'
-,(,)u,,
-,(,)\
.""n(# u,,
-,61)
rt,o
-,(')) tu,.r.
(+,
"
{
" **a {,
*n (r# r,,-,(,) tu,.r.
-, 1')w,
?)}
(r4r, r,1),,0
)
22;e{,""r'
(#r,,t")) tu"n
(#r,,r,1)v,,1,1q,,',] (3.80)
}
Notethat(3.80)is equivalentto (3.31).Thenfor thesamereasons,[/(")]t,r reducessimply
to:
: +I
[1(')]',,
o"
P.:-
(
/2.G
\
/)
G
)
(!!rt,,-,) tu,'n
("!!rt"-,\\ ur,-,ur,-,
I rs.stl
- i _ :tE E { tanh o "
\
/
[
\o'
/
)
which is similar to (3.70) in the caseof BPSK modulation.Thus we obtain the same
expressionfor the stochasticCRLB in caseof MSK andBPSK transmissions
as givenby
8.7r.
62
Bibtiographie
t t l C. R. Rao, "Information and accuracyattainablein the estimationof statisticalparameters,"
Bull. CalcuttaMath. Soc., vol. 37,pp. 8 1-91, 1945.
tzl S.M. Kay, Fundamentalsof StatisticalSignalProcessing: EstimationTheory,Upper Saddle
River, NJ : PrenticeHalL1993.
t3l A. N. D'Andrea,U. Mengali, and R. Reggiannini,"The modifiedCramer-Raoboundand
problems,"IEEE Trans.Comm.,vol.24, pp. 1391-1399,
its applicationsto synchronization
Feb.-Apr.1994.
I4l U. Mengali, and A.N. D'Andrea, SynchronizationTechniques
for Digital Receivers.New
York : Plenum,1997.
tsl H. Steendamand M. Moeneclaey,"Low-SNR limit of the Cramer-Raobound for estimating
the time delayof a PSK, QAM, or PAM Waveform,"IEEE Comm.Letters,vol. 5, pp. 31-33,
Jan.2001.
t6l M. Moeneclaey,"On the true and the modified Cramer-Raoboundsfor the estimationof a
scalarparameterin the presenceof nuisanceparameters,"IEEE Trans.Comm.,vol. 46, pp.
1536-1544,Nov.
1998.
l7l J. Riba, J. Sala,and G. Ydzquez,"Conditional maximum likelihood timing recovery: Estimatorsandbounds",IEEE Trans.Sig.Process.,vol. 49,no. 4, pp. 835-850,2001
digital synchronizationl'SignalProcessingAdt8l G. Vizquez, and J. Riba," Non-data-aided
vancesin WirelessCommunications,EnglewoodCliffs, NJ : Prentice-Hall,2000, vol. 2, ch.
q
[9] I. Bergeland A.J. Weiss,"Cram6r-Raoboundon timing recoveryof linearlymodulatedsignalswith no ISI," IEEE Trans.Comm.,vol. 51, pp.134-641,
Apr. 2003.
"TrueCram6r-Rao
boundfor tiH. Steendam,
andM. Moeneclaey,
tl0] N. Noels,H. Wymeersch,
ming recoveryfrom a bandlimitedlinearly modulatedwaveformwith unknowncarrierphase
and frequency"I EEE Trans.Comm.,v o7.52, pp. 473 -383,Mar. 2004.
of square
andS.Affes,"Cram6r-Rao
boundsfor NDA SNRestimates
tl 1l F.Bellili, A. St6phenne,
QAM modulatedsignals",Proc. of IEEE WCNC'09,Budapest,Hungary,Apr. 5-8, 2009.
63
ll2l F. Bellili, A. St6phenne,and S. Affes, "Cram6r-Raolower boundsfor non-data-aidedSNR
IEEE Trans.Comm.,vol. 58, pp.32llestimatesof squareQAM modulatedtransmissions",
3 2 1 8N
, ov.2010.
'All-digital PAM impulseradio for multiple
access
t13l C.J. Le Martret and G.B. Giannakis,
throughfrequency-selective
multipath," in Proc. IEEE GLOBECOM'2))}, pp.71-81,2000.
|41 M.Z. Win and R.A. Scholtz,"Impulseradio : How it works,"IEEE Comm.Lett., vol. 2, no.
2, pp.36-38,Feb.1998.
64
Chapitre4
A Maximum Likelihood Time Delay
Estimatorin a Multipath Environment
UsingImportanceSampling
1A. Masmoudi,F. Bellili, S. Affes, and A. St6phenne,'A Maximum Likelihood Time
Delay Estimator in a Multipath EnvironmentUsing ImportanceSampling", submitedin
IEEETrans.on Sign.Process.
DOI: 10.1109/TSP.2012.2222402
IEEE (c)
Abstract
Dans cet article, nous pr6sentonsune nouvelle impl6mentationdu critdre de maximum
de vraisemblancepour I'estimation du d6lai de propagationdansun milieu multi-trajet,
puis nous6tendonsla m6thodepropos6epour l'estimationde la diff6rencedu tempsd'arriv6 quand le signal 6mis est inconnu. La nouvelle techniqueimpl6mentele conceptde
l"'importance sampling" (IS) pour trouver le maximum global de la fonction de vraisemblance.Nous 6vitonsla traditionnellerecherchemultidimensionnelleet les m6thodes
it6rativespour maximiser la fonction de vraisemblance.Nous montronspar simulation
que cette m6thodepermetd'estimer desddlaistrbs procheset offre de meilleuresperformancesqueles m6thodessous-optimales
tellesqueMUSIC. Le principal avantagede cette
mdthodeest que la convergenceau maximum global est garanticontrairementaux algorithmesit6ratifs qui d6pendent6troitementde l'initialisation.
In this paper,we presenta new implementationof the maximumlikelihood criterionfor the
estimationof the time delaysin a multipathenvironment,andthen we extendthe proposed
methodto the estimationof the time differenceof arrival when the transmittedsignal is
unknown.The new techniqueimplementsthe conceptof important sampling(IS) to find
the global maximum of the compressedlikelihood function in a modest computational
manner.Thus we avoid the traditional multidimensionalgrid searchor the iterative methods to maximize the compressedlikelihood function. We show by simulationsthat the
new techniqueallows the estimationof very close delaysand surpassessuboptimaltechniquessuchasthe MUSIC algorithm.The main advantageof our methodis the guaranteed
convergenceto the global maximum,contrarily to the populariterativeimplementationof
the maximumlikelihood criterion by the well known expectationmaximizationalgorithm.
4.1 Introduction
Time delayestimationis a well studiedproblemwith applicationsin many areassuchasradar [1], sonar[2], andwirelesscommunicationsystems[3]. Typically,to allow estimation
of the time delay, an a pri,ori known waveform is transmittedthrough a multipath environment,which consistsof severalpropagationpaths,amongwhich the dominantones,
relativelyfew, areconsidered.If the transmittedwaveformis unknown,only the difference
of arrival times can be estimatedfrom the receivedsignalsat multiple separatedsensors
[4]. In what follows,we will treatthe two cases.
Thesetwo time delayestimationproblemshavebeenextensivelystudiedin previousyears
[5-7]. The maximum likelihood (ML) estimatoris well known to be an optimal technique.
For the problem at hand, the likelihood function dependson the time delays and on the
complexchannelcoefficientsmakingits solutionintractablein a closed-form.A direct im66
plementationof this criterion requiresa multidimensionalgrid search,whosecomplexity
increaseswith the number of unknown delays.Therefore,many iterative methods,such
asthe expectationmaximization(EM) algorithm,havebeendevelopedto achievethe well
known Cram6r-Raolower bound (CRLB) at a lower coast.But their performancesare
closely linked to the initialization valuesand their convergencemay take many complex
iterativestepsand therefore,a tread-offmust be found betweencomplexity and accuracy.
Hence,thereis yet a needfor developinga non-grid-search-based
and a non-iterativeML
estimator.Alternatively, sub-optimalmethodsbasedon the eigen-decompositionof the
samplecovariancematrix, which initially gainedmuch interestin the direction of arrival
estimation,werelaterexploitedin the contextof time delayestimation[8-9]. While these
suboptimaltechniquesoffer an attractivereducedcomplexity comparedto the grid search
implementationof the ML criterion,they still sufferfrom heavycomputationstepsdue to
the eigenvaluedecomposition.Moreover,their performancesarerelativelypoor compared
to the ML estimator,especiallyfor closely spaceddelaysand/orfew numbersof samples.
Motivated by thesefacts, we derive,in this paper,a new non-iterativeimplementationof
the ML time delaysestimatorwhich avoidsthe multidimensionalgrid searchby applying :
i) the globalmaximizationtheoremof Pincusproposedin [10] and
ii) a powerful Monte Carlo techniquecalled importancesampling(IS) offering therebyan
efficient tool to find the global maximum of the likelihood function.
Note here that many other traditional Monte Carlo techniques(besidesthe IS method)
can also be successfullyapplied.However,unlike the IS method,they often require a larger numberof realizationsthat are,in addition,usually generatedaccordingto a complex
probability density function (pdf). Hence they appearto be less attractivefor practical
considerations.In this sense,the importancesamplingtechniquelendsitself as a powerful
alternativein which the requiredrealizationsare easily generatedaccordingto a simpler
pdf. Additionally, it offers a way to processthe obtainedrealizationsin a morejudicious
manner [1]. This methodhas indeedbeen applied to the estimationof the direction of
arrival(DOA) [13], the joint DOA-Dopplerfrequency[12] and more recentlyto the estimation of the time delay in the context of modulatedsignalsand a single propagation
path [14]. Basedon theresultsof theseworks,the IS techniquewasshownto dramatically
reducethe computationalcomplexity of the ML estimateswhile still providing high accuracy.
The remainderof this paperis organizedas follows.In sectionII, we presentthe system
model for the active mode (i.e., known transmittedpulse) and derive the corresponding
compressedlikelihood function to be maximized.In sectionIII, we detail the global maximizationmethodappliedto our problem.In sectionIV the importancesamplingtechnique
is describedandthen appliedto the estimationof the time delaysboth in activeandpassive
(unknowntransmittedpulse).Simulationresultsare discussedin sectionV and, finally,
someconcludinsremarksare drawnout in sectionVI.
67
4.2 SystemModel and CompressedLikelihood Function
Consideran a priori, known signalr(l) transmittedthrougha multipathenvironment.The
receivedsignal is a superpositionof multiple delayedreplicasof the known transmitted
waveform;modeledasfollows :
P
\-
u f t ) : ) ZJ
air(t-rn)+w(t),
(4.1)
where P is the total numberof multipath components,u(l) is an additivenoise and a :
for, *r,. . . , ap)r are the unknowncomplexpath gainsresultingfrom scatteringand
fading throughthe propagationmedium.In addition, {r}l:, arethe unknowntime delays
t o b e e s t i m a t e d a n d g a t h e r e d i n t h e v e ct t o: r f r r , r 2 , . . . , r o \ ' . I f F " : 1 / ? i i s t h e
samplingfrequency,the resultingsamples,takenat instances{nT"}{:,
are:
P
air(nT"- r)tw(nT"), n:0, 1,..., N - 1,
a@7): I
(4.2)
i:I
wherel/ standsfor the total numberof availablesamples.
In general,the IS principle is suitablefor the estimationof non-linearparametersin the
generallinear models(GLM) describedas :
(4.3)
a:Q(o)s+u
wherea : laq), a(7"),. . . , y((l/ - 1)7:)]t is the receiveddatavectorwhich depends
linearly on somenuisanceunknown parameterss and non-linearlyon the delays 0. However,in contrastto the single-pathscenarioin [14], the formulation of the input-output
relationshipin (4.2)cannotbe directlytransformedinto a GLM analogousto (4.3).Here,
the receivedsamplesare transformedinto the frequencydomain where the model could
be expressedin a matrix form. In fact, taking the discreteFourier transformof (4.2), we
obtain :
P
Y(k) :
rrl'\
S-
tr/
i'2nkr:
\arX(k)e---n-- +W(k), k:0,
1..... N - 1,
(4.4)
i:l
(DFTs)
where{y(k)}il-t, {X(k)}il;
and{W(k)}I;t *" thediscreteFouriertransforms
of y(nT"), r(nT") andw(nT"), respectively.Then, consideringthis transformation,the
channelcoefficientsvector o and the time delaysr manifestthemselvesas the linear and
non-linearunknownparameters,respectively.Hencewe transformthe basicmodelin (4.2)
into the generalform of (4.3),usinga compactrepresentation
of (4.4)as follows :
Y:Qo(r)a*W,
68
(4.s)
in which y
y(1),. . ., y(N - 1)]t is viewedas the receivedvector,a :
[Y(0),
and the matrix2 A"(t) dependsonly on the unknowndelaysgathered
l"r, or,. . . , c.p]1T
in the vectorr andis given by :
:
o " (t) : [0 "?1) ,Q"( r z) ,.. ., Q"?p) ) ,
(4.6)
with the columns{Q"?o)}l:rbeing definedas :
Q " ( r o ) : [ X ( 0 ) , X ( 1 ) e - ! 3 ; P ,X Q ) e - & # L , . . . , X ( t / - t S e - ' # y .
g.7)
-1)]"are
a n d X : [ X ( 0 ) ,X ( 1 ) , . . . , X ( l / - 1 ) ] t a n d W : [ I 4 l ( 0 )W
, (1),...,W(N
the (.V x 1)-dimensional
vectorscontainingtheDFT coefficientsof samplescorresponding
to the known transmittedpulse and the additivenoisecomponents,respectively.
First, we considerthe active model where, in contrastto the passivemodel treatedlater
in section4.4.3,the transmittedsignalr(f) is known to the receiver.Now, following the
sameargumentsof [15], the likelihood function of the activemodel (4.5) is given by :
A(z.o) xp(y;r.a.):-*.*p{
7T'"
o.'"
- \ft-iD"(z)o)'(v
d.
-iD"(z)c)
} 1o.rl
)
I
by r and
wherep(Y;r,a) is the probabilitydensityfunction (pdf) of Y parameterized
a and a2 is the spectrumpower of the noise.Actually,the ML solution?x,11rsdelined
as the global maximum of the likelihood function in (4.8) with respectto r. However,
this formulationof the likelihood function imposesa joint estimationof z and o which is
computationallyintensive.Therefore,it is of interestto obtain a likelihood function that
dependsonly on r that canbe more easilyhandled.Observingthat A(r, a) is quadratic
with respectto c, we considerthe nuisanceparameter,o, as deterministicbut unknown
andsubstitute,in (4.8),a by the solutiond(z) which maximizesthe log-likelihoodfunction L(r, a) : ln {A(2, a)} for a given r. Indeed,it can be shownthat 6(r) is given
by:
6(r) : (of;(r)o"('))-'a! Q)v
(4.e)
Replacinga in (4.8) by d(r) and omitting the termsthat do not interferein the maximization with respectto r , we obtain the so-calledcompressedlikelihood function of the
systemasfollows :
-'oy
L.('):#tu *"(z)(of 1')o,(')) 1,7v
.
(4.10)
4.3 Global Maximization of the CompressedLikelihood
Function
To find the desiredML estimate,we need to maximize the compressedlikelihood function
in (4.10) over z. Yet, L.(r)
is non linear with respectto r : hence, a closed-form solution
2Notethat we index A"(z) by a to refer to the activemode.
69
seemsanalytically intractable.It is quite common in the current literature to solve this
maximizationproblem in an iterativeway, as an alternativefor the trivial multidimensional grid search.However,iterativeapproachesrequirean initial guess,usuallytakenfrom
the output of anothersuboptimalalgorithm.The iterative quadraticML (IQML) [16], the
simulatedannealingtechnique[17] and the expectationmaximization(EM) algorithm [5],
takenasexamplein our simulation,aresomeof the mostfamousiterativeimplementations
of the ML estimator.Naturally,the performancesof theseiterativealgorithmsdependseverely on the availableinitial guessand may even convergeto local maximum reflecting
estimateswhich arecompletelydifferent from the real valuesof the delays(corresponding
to the globalmaximum).
In this context,the globalmaximizationtheoremproposedby Pincust4.111offersan alternativeto find the global maximum of multidimensionalfunctions,suchas the one at hand
in (4.10). Interestingly,it does not require any initialization and guaranteesthe convergenceto the global maximum.The ideais very simpleandclaimsthat the solutionis given
b y ( 4 . 1 1 :)
r
r - .€xp{ pL"( r ) }dr
i : l i mp-+oo
%.r:1.2.....P.
J.,. . . J.,explpL"(T)j d.r
(4.11)
where-I is theintervalin which thedelaysareconfined.Deliningthepseudo-pdPL'",.o(T),
for somelarge valueof p6, as :
f |
/-\
Lc,pg\t
)
--
W
exp {p61"(z)}
(4.r2)
then,accordingto (4.11),the optimalvalueof 4 is simplygivenby :
o:l,
i:
lrror'",^(r)dr,
L, 2, ..., P.
(4.13)
Intuitively,we can say that, as p6 tendsto infinity, the function LL,rr(") becomesa PdimensionalDirac-deltafunctioncenteredat the locationof the maximumof L.(r). Thus,
the ML estimateis simply obtainedfrom the evaluationof the P-dimensionalintegral in
(4.13).Yet,this is a difficult taskdueto the complexityof the involvedintegrandfunction
L'".ro(.)is apseudo-pdf
L'",*o(.)].
Onesolutionistoexploitthefactthat
and
[thepseudo-pdf
interpretf; as the expectedvalue of ri, the ith elementof a vectorz distributedaccording
pseudo-pdfL|,roO.Therefore,if one is ableto generateR realito the multidimensional
accordingto L'",00(r),it is reasonable
to approximate
the expectedvalueof z usingMonte Carlotechniques[11] asfollows :
zationsof a randomvector,{r*}f:,
P
1+
-\?,
-R
"
3f|*oG)
L'"'
(4.r4)
k:l
is designated
as a pseudo-pdfsinceit hasall the propertiesof a pdf althoughz is not really a
randomvariable.
70
Hence,we substitutethe complex integrationin (4.13) by a simple samplesaverage.
Clearly, as the numberof generatedvalues-Rincreases,the varianceof the samplemean
becomessmaller and? gets closerto the global maximum of the compressedlikelihood
function. Yet a practicalissueremainsas how to easily generaterealizationsaccordingto
L'.,*o(r).The proposedpseudo-pdfis a non-linearfunction of z and needsto operatein
a multidimensionalspace,which is not suitablefor easygenerationof realizations.One
solutionis to approximatethe actualpseudo-pdfby a one-dimensionalfunction andtransposethe problemof generatinga vectorto the generationof P independentvariables,then
resortto the conceptofIS as describedin the next section.
4.4 The Importance Sampling Based Time Delays Estimation
4.4.1 IS Concept
Importancesamplingis a Monte Carlo techniquewhich makesuseof an alternativedistribution (carefully designed)to generaterealizations.It is usually appliedwhen the original
distributiondoesnot havea practicalform, like L'",po(.)given in our problem.
The approachis basedon the following simple observationon the integral involved in
A.r3\:
:
l, Ir"'"'o"o)d'r
I, Ir"tiffn'(r)d'r'
(4.1s)
wherec'(z) is also assumedto haveall the propertiesof a pdf, called normalizedi,mpor.
tancefunction (IF). Then,theleft-handsideof (4.15)is interpretedasthe meanof qffi
when z is generatedaccordingto gt(r) . Unlike L'.,rr(.), it is of interestto chooseg/(r)
to be a simple function of z. Then, we useMonte-Carlomethodsto numericallycompute
the expectationasdonein (4.I4):
* :,rt-:#,
I, I,,tffi n'Q)d,r:
(4.r6)
where27,is now thekth realizationof r accordingto g'(.).
Clearly,the choiceof g'( ) affectsthe estimationperformance.An inappropriatechoiceof
g'(.) may needa largenumberof realizationsr? to reducethe estimationvarianceand result in a highercomputationalcomplexity.Therefore,the valueof -Rdependson how much
g'(.) resemblesL'.,r.(.). tn the idealcase,generations
accordingto g'(.) arethe sameasif
they were generatedaccordingto L'",r'(.). Therefore,ideally,the shapesof the two functions .g'(.) andL'",,,o(.)shouldbe similar to reducethe varianceof the estimatorgivenby
(4.16)[13]. On the otherhand,we shouldkeepin mind that9'(.) hasto be simpleenough
so that realizationscan be easily generated.Thus sometradeoffsare requiredto choosea
7l
functionas simpleas possibleyet similar to Lt",ro(.).In what follows, we will showthat
owing somesimplilicationsof Lt",ro(.),*" can build an appropriatefunctiong'(.) to properly generatevariables.
Now, comingback to the expressionof the actualcompressed
likelihoodfunction(4.10),
makesthe compressedlikelihood function, and
the inversematrix (Oy1"!O"tr))
consequently
the pseirdo-pdfL'",00(l),very non-linearwith respectto r. One can approximateQ{ (r)O.(r)
by a diagonalmatrix to avoid a heavycomputationof the inverse.In
fact,the diagonalelementsof Of;(z)O
aregivenby :
"(r)
1
l/-
\-''-"'''
1 q - o ' ( r ) ( D o ( r ) ) :l r ,Lr I X ( A ) 1 " . I : I . 2 . . . . . P ,
l(6il''
(4.r7)
k:0
and its off-diagonalelementsare :
1V- I
r,-H,
[ ( { P a ' , ( 7 J { P\0r ) ) ] m ; n
__\-
lx (k)l'""o
{
/ ,
a:0
m , fr:
i)r
k(r* - r.)
N
),
I , 2 , . . . , P r n t J7 -.
to.
(4.18)
It is easyto verify, for r^ f rn, that:
[(of (r)o"(r))]^,*< [(o#(')iD"(z))]r;r
(4.r9)
Althoughthis inequalitydoesnot give sufficientconditionto approximateAf;(r)lD"(z)
by a diagonalmatrix, we verify statisticallythat this inequality holds with very high probability for almost all possiblevaluesof the delay differencerm - rn. To that end, we
considerTm;n: rrn - rn as a randomvariableuniformly distributedin4 l-7, 7] and we
defineK(r^,n) the ratio (4.18)l(4.17)asfollows :
K(',,tn):
'tt#--\
Dfl; lx(A')l'""p
{
tH lx(k)l
(4.20)
cumulativedistributionfunctionof K (r*.n)
Then,we plot in Fig. 4.1thecomplementary
(randomizedaccordingto r*,n), to verify that the diagonalelementsof Of;(z)O,(r) are
indeeddominant,with very high probability,comparedto its off-diagonalelements.Therefore,we adoptthe following well-justifiedapproximation:
N
= ( t lx(k)f) r,,
o! 1r)o,1r)
(4.2r)
where.Iois thep x p identitymatrix.
aWeconsiderherethat the delaysdo not exceeda given real value ? (seesection4.4.2 for furtherdetails
on this assumption).
^0.
H
^0_
!Y0.
^0.
a
-o
0.1
o.2
0_3
0_5
T
0.6
0.7
FtcuRB 4.1 - Complementary
cumulativedistributionfunctionof the ratio K (r,n.n).
Then,we definethe importancefunction,gprO, without normalization,[i.e.,g'(r) :
gr,G)l I go,(u)du] in the activecaseas :
Ho
(,)t\,
"(r)oy
eo,(,)-exp
{r*mY
(4.22)
where p1 is anotherconstantdifferent from p6 for somepracticalreasons.A further discussionon the appropriatechoiceof p6 and p1 is left to the end of this section.
After someeasyalgebraicmanipulations,
we express(4.22)as:
gp,(r):
fr"-o
{r*iryr(',)},
(4.23)
where 1(4) is the periodogram of the data in the frequency domain evaluated at each delay
r; as follows :
l'/
I(-.\
-
t t1o;.t(k)e"p{
i)n
(k,A/
*)l
(4.24)
Now, we commenton the advantageof this choice in (4.23) for the importancefunction
(IF). First, we noticethat thejoint contributionof the different delaysir gr, (.) is separable
into the productof their individual contributionsasseenfrom (4.23).Hence,we substitute
the brute generationof realizationsof the vector z accordingto a multi-dimensionalpdf
to the generationof P independentscalarrealizations(i.e., one realizationfor eachentry
of z) usingthe elementaryIF, I orjr). definedas :
'"P{rr#Xr*rr I(")\
I pr\rt' ) :
.LexP
{t:*mot
r@} ar
(4.2s)
Note herethat,the multiplicativetermsX (k), k : 7, 2,. . ., l/ act as weightingfactors.
They attenuatethe contribution of the frequencieswith low energy in the computation
of I (r;) and henceemphasizethe high-SNR frequencies.In fact, this property improves
considerablythe performanceof the estimatorcomparedto someother approacheswhere
the receivedsignal is divided, in the frequencydomain,by the DFT of the known transmitted waveform [8]. Actually, this operationis not suitablefor narrowbandsignalssince
it resultsin someharmful effectsby amplifying the additivenoise in the low-energyfrequencies.It is suitableonly for widebandsignals,in contrastto our algorithm which is
also adaptedto narrowbandsignals.
Finally, the normalizedIF is given by :
fllr exp{prl (ro)}
g'o'r(t) :
I,
[rf]Lr exp{p'rI(u)}dui
(4.26)
with
l- v'
oryy--r14ry'
Pt
(4.27)
We mention that the choice of the parametersp\ and po arc of great importancesince
its affectsthe performanceof the new estimator.In fact, as alreadymentioned,g'o;(r) is
separable
as the productof P elementaryIFs, A oll), corresponding
to eachdelay16(i.e.,
t
/
r
nP
/
\\
rY
o'r1?): fllr lri?.iD. Hence,in practice,we usethe samesr,,(") P timesto generate
th; P elementsof the vectorz. Actually,for a noise-freeobservation,
the functiongor(.)
exhibits exactly P lobes centeredat the locations of the true delays and at each run, a
realizationis generatedfrom the vicinity of one of the P lobes.However,in the presence
of additivenoise,other secondarylobesappearand ultimately affect the generatedvalues.
For this reason,the parameterpi shouldbe increasedto renderthe objectivefunction gr;( )
morepeakedaroundthe actualdelays{ro}l:r. This behavioris illustratedin Fig. 4.2where
we plot the functiongo,rO for two valuesof p'r.
Yet, we observethatpt,cannotbe increasedindefinitely.In fact, very largevaluesof p',
will ultimately destroysomeuseful lobesand so usefulrealizationsmay not be generated.
Obviously,properchoice of p', is of greatimportance.Its optimal value is the highestone
thatmakesat leastsP main lobesappearin gr,r(.).Moreover,by attenuating
the secondary
lobes,we reducethe probability of generatingundesiredrealizations.Consequentlya good
choiceof p/, reducesthe numberof necessaryrealizations-Rand hencethe complexity of
the estimator.
Recallthat the normalizedIF g'r,(r) is built upon an approximationof the actualcompressedlikelihoodfunctionwhich resultsin biasedestimatesof the delays,especiallyat low
SNR values.However,we emphasizeherethe fact that this biascanbe reducedby the presenceof the actualcompressed
likelihoodfunctionin theweightingfactorL'",00(r)lg'r, (r)
tA
oi(.) shouldhaveexactly P lobes,but the additivenoise makesother relatively small secondarylobes
appear.
74
0
0
FIcURB4.2 -Plot otq oJ.) at SNR : 10 for (a) pl : 1 and (b) pl : 6.
in (4.16).Thus,we can maximizethe contributionof L',.,r.(.)in the weightingfactorby
choosingp', smallerthanp6.
4.4.2 Time Delays Estimation in Active Systems
The IS-basedestimatorrequiresthe generationof realizationsaccordingto p4 (.) then
evaluatingthe following meanvalues:
,,:+>1uort*ff
'"
(4.28)
vp\
*:t
where4, is the kth generated
vector andrp(i) refersto its ith element.
Roughly speaking,the delays can actually take any positive value, but in practice,they
are confinedin the interval [0, 7] where 7 is any positive real value that can be chosen
high enough6so that ri e 10,?] for i, : 1, 2,. . . , P. Therefore,sincetheparameters
are
boundedfrom below and above,it is more convenientto usethe circular meaninsteadof
the linear meanin (4.28).The advantages
of this operationwill be discussedlater.
To introducethe conceptof circular mean,define a random variable X taking valuesin
the finite interval[0, 1] and denoteby G(X) its pdf. The circularmeanof X is definedas
6In network communications,the delaysare usually confinedin the symbol duration,whereasfor radar
and sonar systems,the symbol duration does not really exist and the observationwindow must be longer
than the largestdelay.
75
[ 1 8 ]:
E"{x}:
}z
fo'"*v{iztrr}G(r)d,r
(4.2e)
wherethe operatorl(.) returnsthe angleof its complex argument.Supposethat we have
a setof 11, . . . , rR generated
via the pdf G(.), thenthe circularmeanin (4.29)is:
E"{x}: *r*f
r:L
"*p{7zn',}
(4.30)
In our time delays estimationproblem, we first normalizethe delaysby 7 to transpose
them into the interval [0,1]. Then we apply directly the circularmeanin (4.30).In this
context,the alternativeformulation of the IS-basedestimatoris given by :
FGn)",.'
,,: #r*
{;r"#},
*
(4.3r)
whereF(rn) is the weightingfactordefinedby :
F(rt) :
L'"tr('\)
(4.32)
I oi\r* )
From the formulationin (4.31), we only needto find the angle of a complexnumber.
Therefore,any positivemultiplicativeterm will not affectthe final result.Thus,the two
strictlypositiveconstanrsI,
e"p{p \l (u6)}d,'u6,
[, exp{psL.(r)}rtr and[,
I rfll
usedin the normalizationof L'",r.(.) andg'r,(z), respectively,
can be dropped.However,
the exponentialterms in both the numeratorand the denominatorof the weighting factor
F(.) may resultin an overflowin the computation.To circumventthis problem,F(.) is
substituted
by F'(.) :
(
P
:\')
/
- pr>,1(rt@))f+.zzt
- d.,U,I?t@)F'(rr)- exp
(oot"{r,)
,?,?t
{nol"('*)
Note from (4.33) that the argumentsof the exponentialterms are either negativeor zero
and that the valuesofthe exponentialcannotexceedone.
Summaryof steps
In the following, we recapitulatethe different stepsfor the direct implementationof the
new algorithm :
1. ComputetheDFT [y(0), y(1),. . . , y(l/ - 1)] of thereceivedsignalsamples.
2. Use the Fourier transform coefficientsto evaluatethe periodogramaccordingto
(4.24).
a
J.
Computethe samplesof the one-dimensionalpdf gp\(.), usedfor the generationof
the requiredrealizations,at K pointsas :
I o1\Tt):
exp{p'r1(4)}
(
s:K
t
rl
I : 7 , 2 , .. . , K ,
rt'
L*:r exp{pil (ralJ
(4.34)
where K is the total numberof points in the interval -/. Note that we substitutethe
integrationin the denominatorof Er, (.) by u summationover the discretepointsof
the intervalJ.
4. Generateone realizationof z using O'piO.To do so, we generaterealizationsaccordingto grr(.) P timesto retrieveone realizationof the P-dimensionalvectorz.
More detailson this point areleft to the Appendix.Repeatthis stepR times.
5. Evaluatethe weightingfactor F'(ro) fori : 1, 2,..., R and computethe circular mean of the generatedvaluesbalancedby the weighting factors to find the
ML estimateof the multiple unknown delays.Note that we must evaluatethe term
poL"(r) - p\Dl":rI(r1(m))
for all generatedvectors{r}L, beforecomputing
F'(re).
4.4.3 Time DelaysEstimationin PassiveSystems
In a passivesystem,the transmittedsignalis consideredto be unknown.In this case,only
the time differenceof arrival (TDOA) can be estimatedfrom multiple receivedsignalsat
spatially separateddestinations[4]. In this section,we assume,without loss of generality, the presenceof two separatedsensors.The receivedsignalsat thesetwo sensorsare
modeledas :
P1
ln(tl:
\-a
)
ZJ
orir(l - h;i) *rui(l),
(4.3s)
a2.ir(t- rz,i)+ w2ft).
(4.36)
\
,
/,\
Pz
azft): )
\-f
t-l
and {a^,}131, for n :
7, 2, are the delaysand the complexgains of
the receivedsignal at the nth sensor and {P"}2^:, are the known numbersof multipath
components.For the sakeof simplicity, supposethat y{t) hasonly one signalcomponent
(Pt : 1). The receivedsignal at this sensoris consideredas a referenceand henceit
where {r,,t)l!,
is assimilatedto a noisy known signal.Then, similarly to (4.4),we expressthe sampled
signals(4.35)and(4.36)in the frequencydomainas :
Y,(k): ,.ylX(k)""0-t!#.\
{
77
+w,&),
(4.37)
P2
Yr(k): t' or,,x1t
l"o {-UP}
* *rtr1.
(4.38)
k:0,1,...,N-1,
where{Y1(k)}Lo , {Yr(k)}{:,, {Wr(k)}fl:o and {Wr(t )}{:o arely' samplesof the Fourier transformof samplesof y1(t), y2(t), w{t) and w2(t), respectively.As mentioned
above,the TDOAs will be estimatedby consideringthe receivedsignalin the first sensor
asa reference.
This simplifiesto theestimationof the P2 delaydifferencesAf) : r2;i-rr;r
foyi :1, 3,. . . , Pz. Therefore,we rewrite(4.38)asfollows :
:\
Y2(k)
P2
i:r
[ -ffi
P*t?))
\nv,{rr)""p
I\
|) + w,1n1,
-'
(4.3e)
go:%,,i:I,2,...,p2,
(4.40)
in which
At;t
Wr(k): WzG)-
pz
r
tg \
Biwl(k)."p iz"r
I
t
(4.4r)
Doing so, we highlight the parametersof interestin the expressionof Y2(k). Moreover,
thereis an analogybetweenthe formulation of the activecasein (4.4) andthe passiveone
in (4.39).More precisely,the major differenceis in the referencesignal (X for the active
caseandYr for the passivecase).
Then, gatheringall the frequencysamples,we obtainthe following matrix representation:
Y, :
:
-1)]t
[ % ( 0 ) ,Y r ( l ) , . . . , Y r ( N
ao(L,)p + wp,
(4.42)
wherethe matrix Ao(A") is function of the TDOAs definedas :
o o ( A , ): [ d o ( l f ) ) ,o o ( L l ' ) ) , . . e. ,o ( L t a ) ) ,
f 'n-no)'l
'n J| ,
l.
o,(^y)l:
lo,,tol, ' i : L , . . . ,
F1(1)exp I -t'"n7'
Yt(N - 1)exp {I -
P2,
I
(4.43)
)l'
j2n( N - 1) AY)
(4.44)
p : l o ' " , c \ 2.2. . . , o ' ' " f ' ,
LO.r.r (lt.r
0r.r J
(4.4s)
A" : [nf), rl'), ..., Llo,)f',
(4.46)
is the vector of the TDOAs of interest.Consideringthesenotations,it turns out that the
estimationof the TDOAs canbe performedusing the samealgorithmdevelopedabovefor
the activesystem.We only haveto substitutethe vectorQ"?) by Qo(L,) and X(k) by
Y1(k) in the expressionof the periodogramin (4.24). The remaining stepsfollow in the
sameway.
Now we rediscussthe estimationproblem when P1 is different from 1. The problemthen
consistsof estimating PtxP2 differentparameters.To that end,we refer againto theresults
of the activecase.P1 x P2valuesaregeneratedaccordin1togpr(.) by substitutingX(k)
Then,the generated
and Y(k) in the expressionof 1(.) UVY1(.) andY2(.),respectively.
valuesare classifiedfrom the smallestto the highestand organizedas follows :
L,,k: talil,Lt'L...,al?)1,
(4.47)
whereeachvector{tf)r}f}ris
formedfrom P2TDOAs.The final stepconsistsof evaluating the following meansasin (4.31):
: ** t+*r r"n)"*o* "#}
^7,{
{t
(4.48)
4.5 Simulation Results
To properly assessthe performanceof our new IS-basedapproach,we comparethe performanceof the proposedmethod to the expectationmaximization(EM) algorithm [5]
as one representativeexampleof the iterative implementationsof the ML criterion and
to the MUSIC algorithm proposedin [4] as one representativeexample of suboptimal
subspace-based
methods.The estimationerror of the three estimatorsis also compared
to the Cram6r-Raolower bound (CRLB) which reflects the theoreticalachievableperformancetaken as a benchmarkfor all the consideredalgorithms.In all simulations,the
transmittedpulse is a chirp signal which is widely usedin radar and sonarapplications.
The numberof snapshotsis setto ly' : 70. We consider3 propagationpathswith closelyspaceddelays f37", 67", 87]]. The multipath gain is assumedto be equal for the three
paths.
First, we studythe influenceof the parametersps andp', onthe estimationperformance
of our new estimator.We verify that thereis no dependenceon p6 as far as it is chosento
be higherthanp\ (seesection4.4).However,asillustratedin Fig. 4,3, p\ affectsseriously
the estimationperformanceof the new IS-basedalgorithm. As alreadymentioned,small
valuesof p', may not reducethe effect of the additive noise involved in gr, (.), while too
large valuesreducethe desiredlobesrevealingthe actualdelaysin{ r,r(.) therebypreventing their generation.Therefore,an appropriatechoiceof p', is necessaryin orderto obtain
near-optimalperformance.We seefrom Fig. 4.3 that for p', taking valuesbetween2 andT,
the performanceis almostthe same,andthus the optimal valueof p', canbe freely selected
from this relatively largerange.
79
468101214
performance
Estimation
as a function
of pl
FrcuRe 4.3 - Estimationperformanceasa functionof p',
Now turning to the comparisonof the different estimators,we recall that the EM and
IS-basedalgorithmsaretwo differentimplementationsof the ML criterion.They arehence
expectedto exhibit the sameperformancesincethey both try to maximize the sameobjective function. Yet, it shouldbe kept in mind that the IS-basedand MUSIC algorithmsdo
not require any initialization while the EM algorithm is iterativein nature.Consequently,
we considerfor the EM algorithmtwo scenariosin which the initial valuesare selectedas
randomvariables,centeredat the real time delaysand having a varianceof aT! and 10(
reflecting,respectively,
relativelyaccurateandlessaccurateinitializations.Fig. 4.4 depicts
the performanceof the threeestimators.As expected,the two ML estimatorsperform better than the MUSIC estimator.However,for less accurateinitialization, the performance
of the EM algorithm deterioratesconsiderablyover the entire SNR range.We seealsothat
while the MUSIC techniqueapproaches
the CRLB only as far as the SNR is sufficiently
high ; the proposedalgorithmperformscloseto the CRLB over the entireSNR range.This
is hardly surprisingsince the IS-basedestimatoris far more accurateimplementationof
the ML criterion.Sameconclusionshold for the passivecase,alsoplottedin Fig. 4.6 for
Pt:7
a n dP r : 3 .
So far, comparisonshavebeenperformedas a function of the SNR. To study the resolution power of the different estimators,we considertwo propagationspaths and vary the
delay separationAr - 11 - 12 at an SNR value of 10 dB. The resultsare shownin Fig.
4.5. Clearly, as the differencebetweenthe delaysis small, the estimateis less accurate
for the threemethods.The two Ml-based estimatorsstill perform betterthan the MUSIC
algorithm.For well spaceddelays,all the methodsperform the same.
Another important point to study is the effect of the signal bandwidthon the estima-
80
MSE oflhe
lst path
MSE ofthe 2ndpath
10
ari
510
sNR [dB]
510
sNR [dBl
mean MSE ofthe thr€e paths
10
sNR [dB]
l0
15
15
sNR [dBl
FtcuRe 4.4 - Estimation perforrnanceof the lS-based,EM ML and the MUSIC-type
algorithmsin an activesystemvs. SNR.
tion performance.In fact, sinceall the derivationsare madein the frequencydomain,the
signalbandwidth(definedin the given examplehereas the differencebetweenthe higher
and the lower frequencyin the chirp signal) is expectedto have an impact on the estimation procedure.Therefore,we comparein Fig. 4.7 the three estimatorsunder different
signalbandwidths.Clearly,the proposedmethodoutperformsthe MUSIC algorithm over
the entire bandwidthrange,althoughthe gap betweenthe two methodsdecreasesas the
bandwidthincreases.Note that the EM algorithmis alsolesssensitiveto bandwidthvariations. Sameresultshold for the passivesystembut the simulationswere not includedfor
the sakeof conciseness.
Now we considerthe caseof time varying channels.While the proposedmethod is
primarily developedunder the assumptionof constantpathsgains,we verify through simulationsthat it is also robust to time variationsand that the IS-basedestimatoroutperforms MUSIC-type methodsover relativelylow Doppler frequency.Nonetheless,the performancesof the two estimatorsdegradeconsiderablyas the Doppler factor increases.In
8l
MSE ofthe
MSE ofthe 2nd path
lst path
51015
sNR [dBl
10
sNR [dB]
15
mean MSE ot the three Daths
MSE ofthe 3rd path
51015
sNR [dBl
FtcuRB 4.5 - Estimationperformanceof the lS-based,EM ML and the MUSIC-type
algorithmsin an passivesystemvs. SNR.
fact, the time variations of the channel coefficients are not taken into account when developing these algorithms, and it was shown in [19] that, in this case,the estimatesbecome
necessarilybiasedT.Note, in this case,that we are no longer able to obtain the estimatesof
the channel coefficientsusing (4.9). It is for this reasonthat the EM algorithm was omitted
in this scenario since it is based, at each iteration, on an estimate of a, which cannot be
performed for time varying channels.
4.6 Conclusion
In this paper,we developeda new implementationof the Ml-based estimatorfor multiple time delaysbasedon the conceptof importancesampling (IS). We consideredthe
7tn
[t9], the effect of the time varying envelopehasbeentreatedin the caseof frequencyestimationwith
the MUSIC and ESPRIT alsorithms.
82
FtcuRs 4.6 - Estimationperformanceof the IS-based,EM ML and the MUSIC-type
algorithmsin an passivesystemas a functionof Ar.
FtcuRB 4.7 - Estimationperformanceof the IS-based,EM ML and the MUSIC-type
algorithmsvs. signalbandwidthat SNR : 10 dB in an activesystem.
two casesof activeandpassivesystems.The new algorithmis far lessexpensivein terms
of computationalcomplexity than the traditional multidimensionalgrid searchmethod.
Moreover,unlike the iterative methods,the IS-basedalgorithm doesnot suffer from initialization drawbacks.It performs well over the entire SNR range since its convergence
to the global maximum of the likelihood function is guaranteed.In addition,it avoidsthe
computationburdenof the eigen-decomposition
operationthat is widely encounteredin
classicalsubspace-based
techniquesin multiple parametersestimation.While thesetraditional methodsperform well evenfor closely separateddelays,at high SNR values,only
83
800
1000 1200
Doppler shift
t400
FtcuRe 4.8 - Estimationperformanceof the IS-basedand the MUSIC-typealgorithms
vs. Dopplershiftat SNR: 10 dB.
the proposedIS-basedtechniqueprovides accurateestimatesat low SNRs and for challenging casesof more closely-spaced
delays.In practice,an appropriatechoice of the
parametersp6 and p'l canbeperformedto further optimizethe estimationperformance.
Appendix
Method to generatethe YectorT
In this appendix,we presentsomepracticalhints to easily generatea singlerealizationof
the vectorz.
-First,define{asadiscreterepresentationof
theinterval[0,7] (i.e.,€:0
:If s:7with
1/s beeinga givenstepfor somes).
- Then,generate1 accordingto gp,r(.)using the inverseprobabilityintegrationmethod.
To do so, considera vector z of randomvariablesuniformly distributedover [0, 1]. Then
find 1 - argmax,6lG(r), whereG(r) if the cumulativedistributionfunctionassociated
t" go;(.) (for moredetails,see[14]).
- Then eliminatethe generatedvalue1 from { so that it cannotbe generatedagain.
- Repeatthe last two stepsP- 1 timesto generater2, Tz,. . . , rp andobtainonerealization
of the P-dimensionalvectorz.
84
Bibliographie
t l l M.I. Skolnik,Introduction to radar systems,third edition, McGraw-Hill, New York : 2001.
l2l G. C. Carter,Coherenceand time delay estimation: An applied tutorialfor research,development,testand evaluationengineers,IEEEPress,New York, 1993.
t3l M. C. Vanderveen,A.-J. van der Veen,and A. Paulraj,"Estimationof multipath parameters
in wirelesscommunications,"IEEE Trans.Sign.Process.,vo7.46, pp. 682-690,March 1998.
[4] F.-X. Ge, D. Shen,Y. Pengand V. O. K. Li, "Super-resolutiontime delay estimationin mulIEEETrans.CircuitsSysr.,vol.54, no. 9,pp.1977-1986,Sep.2007.
tipathenvironments,o'
t5l M. Feder,and E. Weinstein,"Parameterestimationof superimposedsignalsusing the EM
algorithm."IEEE Trans.Acousr.,Speech.
Sign.Process.,vol. 36, pp. 477-489,Apr. 1988.
'Adaptive realizationof a maximum likelihood
t6l P. J. Hahn, V. J. Mathewsand T. D. Tran,
time delayestimator"Conf.on Acoust.,Speech,Sign.Process.,vol. 6, pp.3l2l-3l24,May
1996.
t7l K. Gedalyahu,and Y. C. Eldar, "Time delay estimationfrom low rate samples: A union of
approach,"
IEEETrans.SignalProcess.,vol.58, no, 6, pp. 3017-3031,June2010.
subspaces
'Active high resolution time delay estimationfor large BT sit8l M. Pallas and G. Jourdain,
gnals,"IEEE Trans.Sign.Process.,vol. 39, no. 4, pp.78l-787, Apr. 1991.
l9l A. M. Bruckstein,T. J. Shanand T. Kailath, "The resolutionof overlappingechoes,"IEEE
Dec. 1985.
Trans.Acoust.,Speech,
Sign.Process.,
vol. 33,no.6, pp. 135?-1367,
'A
pp.
tl0l M. Pincus, closedform solutionfor certainprogrammingproblems,"Oper Research,
690-694,1962.
tl1l A. F. M. Smith,and A. Gelfand,"Bayesianstatisticswithouttears: A sampling-resampling
framework,"AmenStatist,vol. 46, pp. 84-88,May 1992.
[2]
H. WangandS. Kay, 'A maximumlikelihoodangle-Doppler
etimatorusingimportancesamplingl'IEEETrans.Aeraspqceand Electro.Syst.,vol. 46, pp. 610-622,Apr. 2010.
'An exactmaximumlikelihoodnarrowbanddirectionof arrivalestimat13l S. Sahaand S. Kay,
torl' IEEETrans.Sign.Process.,
vol.56,pp. 1082-1092,Oct.2008.
85
'A
maximumlikelihood
l14l A. Masmoudi,F. Bellili, S. Affes, andA. St6phenne, non-data-aided
time delay estimatorusing importancesamplingl'IEEE Trans.Sign.Process.,vol. 59, pp.
4505-4515,
Oct.2011.
tl5l L. Ng and Y. Bar-Shalom,"Multisensor multitarget time delay vector estimationl' IEEE
Trans.Acoust.,Speech,Sign.Process.,vol. 34, pp. 669-677,Aug. 1986.
tl6] Y. BreslerandA. Macovski,"Exact maximumlikelihood parameterestimationof superimposedexponentialsignalsin noise,"IEEE Trans.Acoust.,Speech,
Sign.Process.,vol. ASSP-59,
pp. 1081-1089,
Oct.1986.
t17] K. Sharman,"Maximum likelihood parameterestimationby simulatedannealing,"IEEE Int.
Conf.Acoust.,Speech,Sign.Process.,1988,pp.274l-2744,Apr. 1988.
t18l K. V. Mardia, Statisticsof directionaldata, New York : Academic,1972.
'Analysisof MUSIC andESPRITfrequencyestimates
for sinusoidal
l19l O. BessonandP.Stoica,
signalswith lowpassenvelopes,"IEEE Trans.Sign. Process.,vot. 44, no. 9, pp. 2359-2364,
Sep.1996.
86
Chapitre 5
Maximum Likelihood Time Delay
Estimationfor Direct-Sequence
CDMA
Multipath Transmissions
lA. Masmoudi,F. Bellili, and S. Affes, "Maximum LiketihoodTime Delay Estimation
for Direct-Sequence
CDMA Multipath Transmissions",to submitin IEEE Trans.on Sign.
Process.
IEEE (c)
87
Abstract
Dans cet article, nous consid6ronsle problbme de synchronisationtemporellepour le
Direct-Sequence
CDMA (DS-CDMA) dansun canalmulti-trajet.Nous d6rivonsles expressionsanalytiquesde la bornede Cram6r-Raopour l'estimation du retarddansles systbmesuni-porteuse
DS-CDMA. Puisnousd6veloppons
deuxalgorithmesd'estimationbas6ssur le critbre du maximum de vraisemblance.Le premier reprendla m6thodeit6rative
"expectationmaximization" (EM). Le secondalgorithmeimpldmentele critbre du maximum de vraisemblanced'une manidrenon-it6rativeet retournele maximum global de la
fonctionde vraisemblance
en utilisantI'IS. Nous g6n6ralisons
aussiles deux algorithmes
et la borne de Cram6r-Raopour les systbmesCDMA multi-porteuses.Les simulations
montrentque l'algorithme EM estbien appropridpour les systbmesavecun grandnombre
d'antennesalors que l'algorithme IS offre de meilleuresperformancesen pr6senced'un
petit nombred'antennes.
In this paper, we addressthe problem of time delay estimationfrom Direct-Sequence
CDMA (DS-CDMA) multipath transmissions.We derivefor the first time a closed-form
expressionfor the Cramer-Raolower bound (CRLB) of multiple time delay estimationin
(SC)DS-CDMA systems.Thenwe developtwo time delayestimatorsbased
single-carrier
on the ML criterion.The first one is basedon the iterativeexpectationmaximization(EM)
algorithm andprovidesaccurateestimateswhenevera good initial guessof the parameters
is availableat the receiver.The secondapproachimplementsthe ML criterion in a noniterativeway and finds the global maximum of the compressedlikelihood function using
the importancesamplingtechnique.Unlike the EM-basedalgorithm,this non-iterativemethod doesnot require any initial guessof the parametersto be estimated.We also extend
both the SC CRLB and the proposedSC algorithmsto multicarrier (MC)-CDMA systems
by exploiting the frequencygain over subcarriers.In this work, the estimationprocess
canbe performedusing the channelestimateor directly from the receivedsignaland thus
we cover all possiblecases.By an adequateformulation of the problem, we are able to
exploit the time and frequencycorrelationif the channelestimateis used.We show by
simulationsthat the EM-basedalgorithmis suitablefor CDMA systemswith largereceive
antennaaffays whereasthe IS-basedoffersbetterperformancefor small array sizes.
5.L Introduction
The most important challengefor wirelessnetworksis the developmentof robust transceiversthat are ableto transmitat high datarateswith a high bandwidthefflciency.Codedivisionmultiple access(CDMA) systemscan satisfythis requirement.CDMA hasbeen
adoptedas the multiple accessschemefor the third generationcellular mobile systems
88
becauseof its flexibility in cell planning,usercapacity,supportfor differentratesandrobustnessto multipath channel.One of the most important motivationsbehind the use of
CDMA is to increasethe numberof simultaneoususers(usercapacity)with acceptableerror performance.OFDM basedCDMA systems,alsocalledmulticarrier(MC)-CDMA, is a
promisingmultiple accessfor high speedcommunicationsystemdueto robustnessagainst
frequencyselectivefading channeland fully usethe availablebandwidth [1-2]. However,
theperformanceof thesesystemsis closelylinked to synchronization.In the following, we
focus on the CDMA array-receiverwhich hasreceivedmuch interestsequelto the performancepotentialit carries 13,4, 5f. Roughly speaking,the post correlationmodel (PCM)
of the despreaddatapresentsthe signalin an interestingway to apply the researchesdone
estimatorwas initially
in the field of affay processing.
A suboptimalRoot-MUSIC-based
developedin [6] to recoverthe time delayand later relined in [1] to significantlyreduceits
complexity.This paperalso investigatesmultiple time delay estimation,yet in an optimal
way in which the ML criterion is adaptedto the PCM.
The problem of high-resolutionparametersestimationhasbeenextensivelystudiedin the
past.In this context,it is well known that the ML techniquealwaysoutperformsthe other
sub-optimalmethodsin the challengingcasesof low signal-to-noiseratio (SNR) values
or small numberof availabledatasnapshots.However,a direct implementationof the ML
criterion requiresa multi-dimensionalgrid searchwhich is of courseimpractical.Alternamethods(which reducethe problem to one-dimensionalgrid
tively, eigen-decomposition
search)[7, 8] haveattractedmuch interestdue to their simplicity and their high-resolution
capacity.Yet, they aremainly basedon the samplecovariancematrix andrequiretherefore
a large numberof datasnapshots.However,as we will seelater,the numberof snapshots
in CDMA systemsis equivalentto the number of receiving antennaelements.Thus applying a traditional suboptimaltechniquewould require a very large numberof receiving
antennabrancheswhich is also impractical.Consequently,thereis a needto derivean efficient implementationof the Ml-based estimatorthat avoidsthe trivial multidimensional
grid searchapproach.To that end,iterativemethodsareusuallyenvisionedto find the ML
estimatessincea closed-formsolutionis, in mostcases,deemedintractable.And to properly evaluatethe performanceof theseestimators,we derivein the first part of this paper
a closed-formexpressionof the Crambr-Rao
lower bound(CRLB).
Motivatedby thesefacts,we developin the secondpart of this paperan efficientschemefor
the estimationof the delaysbasedon the expectationmaximization(EM) algorithm.While
this methodis widely usedin multiple parametersestimation,especiallyfor the estimation
of multiple time delay from an incoming waveform[9], it hasnot beenyet adaptedto the
contextof CDMA systems.In few words,the EM methodoffers an interestingway to decomposethe observedsignalinto different replicas,eachone coming from one path, and
Therefore,the multidimensionaloptimizationtaskis
thentreateachcomponentseparately.
interestinglytransformedinto multiple one-dimensionaloptimizationproblems,resulting
89
in tremendousnumericaladvantages.
Under good initialization, the likelihood function of
the estimatedparametersis increasedin eachiteration and hencethe algorithm converges
to its global maximum.
Alternatively,when a good initialization is not available,we resort to the conceptof importancesampling (IS) to derive, in the third pat of this paper,anothertechniquethat
finds the global maximum of the likelihood function, in a non-iterativeway. In our case,
the likelihood function dependson the time delay and the channelcovariancematrix. To
obtain a function that dependson the unknown delaysonly, the channelcovariancematrix is replacedby its ML estimate(a function of the delaysthemselves).The resulting
objective function, called compressedlikelihood function, is then maximized with respect to the unknown time delay. To that end, we use the global maximization theorem
introducedin [10] that providesan efficienttool of Iinding the global maximumof multidimensionalfunctions.However,it still requiresthe computationof a multidimensional
integralwhich is itself difficult to perform.Yet, this integralcan alwaysbe tackledempirically usingMonte Carlo (MC) methods[11]. Among theseMC methods,the importance
samplingtechnique,in particular,has been shownto be a powerful methodthat reduces
considerablythe computationalcomplexity.Typically, it was successfullyappliedto the
estimationof directionof arrival(DOA) [ 12], thejoint DOA-Dopplerfrequencyestimation
[13] and,more recently,to the estimationof the time delay in the contextof a singlepath
and linearly-modulated
signals[14]. Comparedwith the EM ML estimator,this method
doesnot needany initialization and doesnot suffer from any convergenceproblem,yet it
is computationallymore intensive.The contributionsof this work coverSC-CDMA and
MC-CDMA aswell. In the estimationprocess,we distinguishtwo major cases.In the first
one,we estimatethe delay directly from the receivedsignalover the antennaarray.In this
case,we provethat time, frequencyand spacedimensionsaremixed togetherto obtainone
dimensionwhich is the productof the three.On the otherhand,a channelestimatecan be
obtainedprior to time delay estimation[26] then we usethe channelestimateto estimate
the delays.
This paperis organizedas follows.In sectionII, we briefly introducethe post-correlation
model. Then, in sectionIII, The analyticalexpressionof the CRLB is derived.In section
IV, we developthe new EM-basedML time delay estimator.In sectionY derivationdeIn sectionVI, we extendthe presented
tails of the new IS-basedestimatorare discussed.
work to the caseof multi-carrier (MC)-CDMA. SectionVII presentssomesimulationresults that corroborateour findinss and finallv someconcludineremarksare drawn out in
sectionVIII.
90
5.2 SystemModel and Background
We considera CDMA communicationsystemwherethe receiveris equippedwith M receiving antennaelementsthat capturesignalstravelling through a multipath propagation
environmentconsistingof P different paths.The signalsreceivedon the M antennasare
uncorrelatedwith the spreadingcode and sampledat the chip rate Q. Denoting the processinggain by L (i.e.,L : T lT" with 7 beingthe symbolduration),the resultingpostcorrelationdataof the spatio-temporalobservationof the nth receivedsymbol is modeled
by the following matrix form [6] :
Zn:
G,TnDT (r)s, * ly',,
(s.1)
where sn: b,{, is a function of the unknowntransmittedsymbol bn andthe squareroot,
is the (M x L)-dimensional post-correlation
$,, of the total receivedpower r!2"and-Ay',,
noisematrix. The pth column of the matrix D(r) that gathersthe time delay parameters,
T1-,72,.. . , Tp,is givenby :
d, : lp"(-rr), p.(7" - ro),.. ., p"((L - 1)7" - ro))',
(s.2)
wherep"(.) is the correlationfunction of the spreadingcode.Gn is the M x P spatial
propagationmatrix and T," is a P x P diagonalmatrix representingthe normalizedpower
ratiosoverthe differentpaths[i.e.,trace(Tl : t)] [6]. Thesetwo matricescanbe further
gatheredin one singlespatial-response
matrix Jn (i.e.,Jn:
GnT",) andby includingthe
scalarterm sn in the matrix2Jn, amore compactform of Zn is givenby :
zr:D(r)fi
+lr;.
(5.3)
in (5.3),the original problemcanbe interpretedasthe estimation
Using the representation
observedon.L antenna
of the time delay,involvedin the matrix D(r),from M snapshots
branches.Each column of ZI representsan observationvector and the columns of $
are interpretedas the transmittedsignalsfrom P different sources.If we supposethat the
delay vectorz remainsconstantover N transmittedsymbols,a compactrepresentationof
(5.3)over l/ symbolsis givenby :
Z:[ 2T,2T,...,ZK]
: D(r)Jr + Ar7,
(5.4)
: [AIrr,N{,. .. ,lffr].
withJr : lJT, $,..., JK]and.Alr
Usually, high resolutionmethods(when applied to time delay estimation)transformthe
2Forthe sakeof simplicity, we keepthe samenotationJn for Jns,. Hence,pleasenotethat the following
fbrmulation holds, unlessspecifledotherwise,lbr both data-aided(i.e., s, is a known referencesignal) and
transmissions.
non-data-aided
9l
problem into the frequencydomain in order to obtain a formulation that is similar to the
one encountered
in frequencyestimationl9l, |71, [18], after which high-resolutionmethods,suchas Root-MUSICcan be appliedto estimatethe delays(asin [6]). Following
the samelogic, we perform a column-by-columnfast Fourier transform (FFT) of ZT to
obtain:
(s.s)
Z:D(r)Jr+N,
where,A/ is the resulting transformednoise matrix and D(t)
dependsonly on the unk-
nown delaysand is given by :
D(r) : ld(r1), d(rr),. .. , d(rp)1,
wherethe columns{d(4)}f:,
(s.6)
aregivenby :
d\Tp):lCg,C1e
_i2n(L-t)rp,-
_j2nrp
L
,...,CL-r€
"
l',
(s.7)
and{c}l:oI arethe FFT coefficientsof the spreadingcodecorrelationfunction. Note that,
for CDMA systems,the correlationfunction of a perfect spreadingcode is a Dirac function, and hencethe corresponding
FFT coefficientsare constantin this ideal case.This
featureholds true as a very good approximationevenwith practical spreadingcodes[1],
t6l.
Before exposingthe main contributionsof this paper,we mentionthat the following developmentis applicableon the estimateof the channelcoefficientsmatrix. Actually,consideringagainthe formulationin (5.1),Zn canbe written asfollows :
Zn:
I:lnsn * Nr,
(s.8)
in which Hn :
J"Dr (r) denotesthe overall spatio-temporalpropagationmatrix. Interestingly,the model in (5.8) can be usedto estimatethe channelresponse,II,, rn an
eflicient way. One can useany blind channelestimatorto obtain an estimate, fin, of LIn.
More detailson this subjectcan be found in [6]. Now, taking into accountthe estimation
writtenas :
"nor,frnis
fr|:D(r)fi+EI,
(s.e)
where E[. is the correspondingchannelerror matrix. The varianceof the entriesof EI
dependsofcourse on the noisevariancein the receivedsignal[15] and it hasbeenstated
that the power of En is lower than the power of .A[,, which has the effect of increasing
the SNR. Clearly,time delay could be estimatedform the column-by-columnFFT of Ff,
in (5.9) as well as from (5.5),with the only differencethat the noisepoweris reducedin
(5.9).In the remainof this paper,we considerthe formulationin (5.5)sincean estimateof
the channelis not alwaysavailable.
92
5.3 The CramBr-RaoLower Bound
Beforewe dealwith the two estimators,we derivein this sectiona closed-formexpression
for the CRLB for the problem at hand, which will be used as a benchmark,in addition
to the Root-MUSIC algorithm, againstwhich we evaluatethe performanceof the new
estimators.
In fact,the Cramdr-Raolower Bound (CRLB) is a well known lower boundfor the variance
of unbiasedestimatorsof an intendedparameter.Many works have so far dealt with the
evaluationof the CRLB for the time delay estimationproblem but, as far as we know, no
contributionshavebeenmadeyet in the contextof multipathtime delay estimationin DSCDMA systems.To that end, we assumethat the multipath fading coefficients,gathered
in $, are random variableswith unknown covariancematrix fily. Therefore,the vector
of unknownparametersinvolved in the estimationprocessis :
o : lr, W{-Ry(m,n)}'*.,:r,3{.R;(rn, n)}o*.,:r, otl'
(s.10)
with z :
lrt, r2t. . . , rp)r. In the following, we supposethat the differentcolumns
of Z, denoted26 arremutually independentand the columns of A/ are also mutually
independentand Gaussiandistributed.Under theseassumptions,
the probabilitydensity
function (pdf) of Z, parameterized
by r andl?"r (the covariancematrix of the columnsof
$1, is givenby :
I
B 1 Z :r, R t):;frfr1
-'
tI-
(det(2(z),RtD(r)'
(MN
""ot-?tr
(o1r1aro1ryH
+ o2Ir.\
rrMN
* o2I7 ) )
-t
zr\ ,
( 5 . 11 )
w h e r e d e t ( .r )e t u r n s t h e d e t e r m i n a n t o f a g i v e n m a t r i x a n d r : l r r , T 2 , . . . , r p ) r . T h e n
the log-likelihoodfunction,L(r, Rr) : ln(F(Z;r, Rt)), reducessimplyto :
L(r, Rr): - ln (det(2(z) RroH (r) + o2t"))
-
't
r
#
MN
\--
(') + o'rr)-'zn'
F-",(o1r1nrou
/
6'12)
The entries of the Fisher Information matrix (FIM), denoted here 1. are given by3 :
I(m,n): Mtrace
,
{yor'!3=or'Pz=\
-"' )a(m)-"' )a(n) )'
(s.r3)
with Rs being the covariancematrix of 26 givenby :
Rz : D(r)RrDH (r) + o2r7.
'Note that the CRLB of the joint estimationof all parametersis simply the inverseof the FIM .I.
93
(s.14)
However,the derivationof the FIM startingfor (5.13) appearsto be intractable.Alternatively, the CRLB is asymptoticallyequivalentto the error covariancematrix, Cy1(z) :
E{(? - ,)(? - ,)'}, of the maximumlikelihoodestimate,as M tendsto infinity [25],
which meansthat :
Cun(r):
(s.1s)
CRLB(z)'
Recallthat the ML estimatei of r verifiesthe following equation:
u9@:0,
(5.16)
OT
where AL(.)lAr is the gradientof I(.) with respectto z. Applying the Taylor series
expansionto the left-handsideof (5.16)andkeepingonly the first two termsof this developmentleadsto :
} L ( r , P " 1 ) , 0--2 L 1 r . P - t ) , . . - \ n
-t
t) : g,
+
d"--=t
at(i
(5.r7)
whereA2L(?, P-r)I 0r2 is a Hessianmatrix.Hence,from (5.17)we obtain:
| 02L6(r,,Rr)'l
-t
0 L(r, P"1)
(s.18)
rmrnl
lOr")OT
where02L6(r, R) l0r2 is a Hessianmatrix whenfr.2 -+ Rz and,6-+ o as M tendsto
oo. It follows that :
a) 0L(r.Rt)'})
_ l02roc.Rt)l-' (.fT tr
(]^",
vML-t
[r,r,(,,.,,1-lr.,n,
{aw.
or
or
ar, J
a", J \;::"1I)l
The analyticalexpressions
of thegradient0L(r, P"r)I Er andthematrix02Lo(r, P'1)I 0r2
(derivedin [25]) leadsto the following analyticalexpressionfor the CRLB of time delay
estimates:
cRLB(r):
n2 | r .
nl (u"
;N ' LI \
-)l-t
[1- il] u) x (n1nH1r1n;'ot")nr)' | | ,6.201
/_l
where* standsfor the element-wiseproduct,fI is an orthogonalprojectormatrix defined
asfI : D(t) (OH 1r1o1t))-t o' (r) andthematrixU is definedasfollows:
U:lur,
ui:
? . 1 2 , . . u. ,p ] ,
0d(ra)
----;-.
OT;
94
(s.2r)
(s.22)
5.4 ExpectationMaximizationAlgorithm
The EM algorithm is a computationalmodest method to find the maximum likelihood
estimatewhen a closed-formsolutionof this one is intractable.Rewritethe loe-likelihood
function in (5.12) in a more compactform as follow :
L(r, F-r): - ln (det(R2)) - trace
{n2'fr"}
,
6.23)
whereR"b"ingan estimateof R2 computedfrom the columnsof Z asfollows :
't
I;L g :
MN
r
(s.24)
f-zzH
u) . Z i Z i
^
^tt\
Note herethat the log-likelihoodfunction,L(r,l?;r), dependson the delaysvectorr of
interestand the covariancematrix -R7. Hencethe problem can be formulatedas follows :
maximizeL(r, R) with respectto r andRr. We alsomentionthat while o2 is usually
unknown,it canbe easilyestimatedeitherby averagingtheL - M smallesteigen-values
of Rz or simply by exploiting the estimatedpower carriedout in a previousstageof the
receiver[6].
Unfortunately,the aboveexpressionof the likelihood function cannot lead to a closedform solutionfor its maxima.Thus,we resortas a first option to a well-knowniterative
algorithm, namely the EM algorithm [16], to resolvethis problem numerically.The purposeis to decomposethe observation,{Zu}Yl
then estimatethe
into P complete-data,
delays separatelyfrom each complete-data.This is equivalentto performing P parallel
the commaximizationsovera one-dimensional
space.This methodreducesconsiderably
putationalcomplexity comparedto the brute grid searchsolution. For this purpose,we
deflnethe setof completedataas :
. t n ) ( i ): J r ( i , , p ) d ( r r +
) n@(i),
p : 7, 2,..., P, i : 7, 2,..., MN.
(5.25)
snapshotfrom
wherez(d (i) canbe seenas the receivedsignalon the i'h spatio-temporal
thepth path.From (5.25),the covarian o, .(d (i), n {ztol (i)z@ ('i)H} is givenby :
""
R"{o): e}d'(ro)d(rr)u
*
i,
",
(s.26)
with {e2"}l:, beingthe diagonalelementsof Rr andn1(k) is an arbitrarydecomposition
of the estimationerror(i.e.,No : D;:rn@ OD.From (5.25),any column,2i, of 26 can
be written as a function of the completedataas follows :
D
Z ;- :
\i:
tnl z.r
z'''lll.
)
.Ll
p:L
95
(5.27)
Now, we arein a positionto describethe Expectation-step(Z-step) andthe Maximizationstep(,4,I-step)
of the EM algorithm.The E-stepconsistsin findingthe conditionalexpectationsof the samplecovariancematrices{fr"rr}f:,
of the completedatadefinedas :
fr'^r:#
Given Rf-'l
(s.28)
np6r{c-r} (thepreviousestimatesof Rr andr at iteration(q - 1) andfr.2,
the expectationof R"rnl canbe computedfrom the classicalformulas of the conditional
expectationwith Gaussiandistributedrandomvectorsto yield :
u {fi^r;fr";rr9-'}'
fiL1L:
'{c-r}}
: nLnl,
Rt;!,
("9,)-' R" ("9t)-' ^L'J,*t"II,("9,)-' *l?],,6.2s)
where the matricet Rgj are computedat each iteration from the estimates,jn-t]} unO
,z{o-t} computedin the previous iteration. Now, turning to the estimationof Rl), the
procedureis differentfrom the one usedin previousEM algorithmsin [9] and [19] to estimatethe covariancematrix. In fact, in [9] and [19], the covariancematrix of the received
signal is simply diagonal,which is not the casein our work. Therefore,we resortto another approachto estimatethe covariancematrix Rl). first. supposetnatRtj is a Toeplitz
matrixa (caseof stationaryprocesses).We adoptthe methodproposedin l2ll @riefly detailed next) to the estimationof Toeplitz covariancemaftices,which is also basedon the
EM algorithm making it well suitedto our algorithm. Then, we define the l/" x l/" circulant extendedversionof Rs, denotedas -R". The matrix fi|" representsthe covariance
matrix of the extendedvectors {Zr}Y{, where Zi consistsof the vector Za augmented
bV (N" - l)-dimensional null vectors.The covariancematrix .R" is characterizedby its
eigenvaluesas follows :
-R" : FH RsF,
(s.30)
where lf is the standardAL x l/" discreteFourier transform (DFT) matrix and -R6 is a
diagonalmatrix constructedfrom the eigenvaluesof .R". The DFT transformof 26 results
in the rotatedvectorsCt : FZn foli : 7, 2,. . . , M N. Denotingby fr" the estimateof
-R6rusing{Cn}{:, (i.e.,-0" : lllvl N Dyi CoCf), theexpectation
of -R6rconditioned
on R7 andRz
applyingthe sameformula usedto find (5.29) is given by :
E(AdRz,R) : n"" (R"'fi-" IRZ')' - R"') Rzc + Rc.
(s.31)
whereRss is the cross-covariance
of C; and Z;. Noting thatZ; : F'C* with F :
FII L OITand 0 is the (.L x ,n/"- I) null matrix, the cross-covariance
matrix R2s is equaT
aThis assumptionimplies that the covariancebetweenZ^ and Zn dependsonly on the differencebetweenm and n, correspondingthereforeto stationaryprocesses.
96
I
rc RcF. Thenthe estimateof Rs at iterationq is givenby :
- @y-rjI ') F"nf-'r + R\-,tlS.:zl
Rgj :oiag(n$-'\ r (tng-tj) LAz(Rr;-,t)-,
and,R2 is obtainedusing the transformationZa : FH Cuas follows :
Rt;j : r" ngt,
(s.33)
Finally,it hasbeenshownin [21] that the stablepoint of (5.33)is equalto the maximum
likelihoodestimateof Rs,.
During the M-step of the EM algorithm,we aim to maximize the log-likelihood function
of the complete-data
with respectto the parameters
of interest{ro},!:r.It is the sameobjective function given in (5.I2), with the true expectationof the completedatareplacedby
theconditionalexpectationof ,tn (l; in otherwordsfi|s substituted
by R"ws arnd
R."by
fr.tt . Thus we obtain the log-likelihood function of the complete-data,Lr(r, R,ro>),as
follows :
- trace
Lo(ro,R.<o)): - ln (det(,R,1";))
{e'":},^;e,}
(s.34)
j
Then,at iterationq, theestimaterlq andR,61are thosewhichjointly maximize Lo(re,R"rot).
Usingthe eigen-decompositions
of R"{d, the log-likelihoodfunctionof the complete-data
canbe expressed
as :
L , ( r o , R " , o: , ) - m ( 1 7 + * ) - U r - 1 ) l n( + \ r/
\.
\r/
(
P
/^r-r\
P\
-;)a@)'RL']'a(d3t'""'
(s3s)
(€j1]')
(4*g
r
which emphasizesthe dependenceof Lr(ro, R"r,t) on r, and e2r.fne closed-form expression of its maximum with respectto el, for a given to{n}.i, ,
: a1r{u}1H
,"o{n}
frL1,1,a6'})
+
(s.36)
Now, injectingthis expressionin (5.35)yields the following one-dimensional
maximization problem :
:arsm€x-m
rj."}
fr!n!,a1,l\
{a1ro;"
fiafr,l'frln},a1ro), 6.37)
or simply the problemof maximizingd,(r)H R{oJ,d(rr). This follows immediatelyfrom
the fact that the functionf (r) : - ln(r) t P f o2r is monotonic.
So far, the ML estimatehas been found in an iterativeway. However,this methodneeds
an initial guessof the parametersfrom which the algorithm startsoperating.Alternatively,
to avoidall initializationhurdlesandissues,we developin the next sectiona non-iterative
algorithmto find the ML estimateswithout grid searchbasedon importancesampling.
sNoting that the matix e2rd(r)d(tr)H is of rank one with -L - 1 null eivenvaluesand one equal to ef
,
the eigen-decomposition
of R"1o1can be easily done.
5.5 The Importance Sampling Technique
Similar to the abovealgorithm,we startfrom the expressionof the log-likelihood function
in (5.12).A directmaximizationof this functionimposesjoint maximizationover r and
.R.7'.Therefore,it will be of interestto formulate an objectivefunction that dependson
the time delayonly.To that end,we first maximizethelikelihoodfunctionwith respectto
the nuisanceparameters.R;. For this purpose,it can be shownthat the value of Rr that
maximizesL(r, P-t) for a fixedvectorr is :
Ay' : (oH1r1o1'))-'oHt)fr.2o(r) (DH(r)o(r))-' - o' (oH1r1o1'))-'.
(s.38)
Then,injectingRy" i" (5.12)yieldsthe so-calledcompressed
likelihoodfunctionof the
system:
L.(') : \rru"" ("e") - m (a"t(nfr,n + o21rr-il))) ,
in (5.20),is definedas fI : D(r) (DH (r)D(r))
wherefI, introduced
(s.3e)
2H(z). Now,
the maximumlikelihood estimatesof the time delay areobtainedby maximizing the obtained compressedlikelihood function (again,the most obviousoptimizationtechniquethat
naturally comesto mind is to perform a P-dimensionalgrid search,whose complexity
increaseswith the numberof delays)L"(r) with respecttor.In this section,as an alternativeto the iterativemethodalreadypresentedin section5.4, we implementherethe ML
criterion in a non-iterativeway. We resort to the global maximizationtheoremof Pincus
[10] in order to find the global maximumof the multi-dimensionalfunction at hand.In
fact, accordingto [10], the globalmaximumof L"(r) with respectto r is givenby :
?-:
'P
IL^'oexPr{qLJ')}dr
Irexp{pL.(r)} dr
lim [t^
;';; [,
(s.40)
with J : [0,7] beingthe intervalin which the unknowndelaysare supposedto be confined.Clearly.asp tendsto infinity.thefractionL"+X*#
becomesa multidimenJJ... JJexplptJclT)jl
sionalDirac function,centeredat the global maximumof L.(.). Therefore,if we define
Ii.,(.) as :
thepseudo-pdf
L'"o(r) :
exp{pL"(r)}
I,
I r exp{pL.(r)} dr'
(s.41)
theML estimateof {ro}!:r, obtainedby applying(5.40),canbe reformulatedas :
tt
J,
'i:1, 2, ..., P,
Jr'oL'''oQ)dr'
(s.42)
later).The funcwhereps is a sufficientlylargenumber(whoseoptimalvalueis discussed
tion L'",.o(.)is calledpseudo-pdfsinceit hasall the propertiesof a pdf, althoughr is not
98
truly a randomvariable.We note from (5.42)that the ML estimaterequiresthe evaluation
of the multi-dimensionalintegral,which is usually diflicult to perform in practice.However, exploitingthe fact that L'",r.(.) is a pseudo-pdf,the involvedintegralcan be simply
interpretedas the mean value of ro, when the hole vector r is distributedaccordingto
L'",ro(.).Therefore,
one caneasilyevaluatethis mean- andhencethe integralsin (5.42)
- in orderto obtain? : lit, ?2,. . . , 7")' usingMonte Carlotechniques[11] :
p
1+
E
- " )-,'r"'
(s.43)
k:r
where {"r}*:, areR realizationsof r, with z being distributedaccordingto L'",.o(.).
But anotherproblemariseshere: how to jointly generate{ro$:, for a multidimensional
likelihood
randomvariable.Actually.L'",ro(.)is constructedusingthe actualcompressed
functionin (5.39),which is a multi-dimensionalfunction; makingthe generationof the
vector r a very difficult task if not impossible.Therefore,it is of interestto find another
pseudo-pdfto generatetherealizationsinsteadof usingL'r,ro(.).To do so,we resortto the
conceptof IS as detailedbelow.
First, we mentionthat IS is a powerful Monte Carlo techniquel23l which allows generatrngrealizationsusing anotherdistributionthat is simplerthan the actualone.Throughthe
IS technique,the generatedsamplesare weightedand averagedin a judicious mannerto
obtainthe desiredestimates.This efficientweighting operationimprovesconsiderablythe
performanceachievedby the IS methodcomparedto other Monte Carlo techniques.The
IS approachis basedon the following simple observation:
I, . l,rr"lffis'(r)d,r, (s.44)
I l,f {,)r'.,,o?)d,,:
where g'(.) is anotherpseudo-pdfcalled normalizedimportancefunction (IF), whose
choiceis discussedlater and /(.) is any given parametertransformation.
Now, the prowith respectto the
blem is recastas the computationof the expectationof f ft)ffi
distributiong'(.) ; which is simply performedvia Monte Carlo methodsas follows :
* : r?,)L+#,
I, l;t)+#s'e)rtr=
(s.4s)
in which the realizations{26}[1 arenow generatedaccordingto g'(.). Yet, a greatattention shouldbe given to the choiceof g'( ). In fact, the accuracyof this methoddepends
on the similarityof the shapesof L'",00(.)and S'(.).In the bestcases,the global maxima
of L'.,ro(.) andg'(.) are the same.Still L',,,.o(.)is a complicatedfunction of r andg'(.)
must be as simpleaspossibleto easilygeneratethe requiredrealizations.Therefore,some
trade-offsmust be found in the constructionof the importancefunction. Moreover,an appropriatechoice of g'(.) reducesthe number R of realizationssincethe generatedvalues
99
will appearas if they were generatedaccordingto the original pseudo-pdfL!,ro(.) when
g'(.) is faithful to L'",ro(.).Next, we discussthe appropriatechoiceof g'(.).
First, as an alternativeto the actualmultidimensionalcompressedlikelihood function, the
importancefunction should be a separablefunction in the different delays, {ro}l:r, to
reducethe generationof a P-dimensionalrandomvectorto the generationof P scalarrandom variables.We thereforesimplify the expressionof the compressedlikelihood function
L"(.) to lind g'(.). Indeed,it is seenfrom (5.39) that L"(.) involvesthe sum of two independentterms.Thesetwo termscan be written as functionsof the eigenvaluesot[Ifr.zft
as follows :
\t-e
[^")
(3)
P
:f
rn
i.:L
I Llno2,
(s.46)
and
7
r
r
1
/-=
: -trace lrJBzfll
;trace (IJP.z)
,/
\
\
where)r, )2, . . . , )p arethe eigenvalues
of fIRzfI.
. =,
_$)o
./
-o,oo
(s.47)
".
Clearly,the term Lf:r)
is doml-
in (5.39)we droptheterm LIno2,
"*requently,
independent
on the delays.in In(det (nfi."tl+
o2(1"and it is reasonable
ne"r)
glectthetermh (det(nR zn + 02(r L with respectto jtrace ("A"). Moreo")))
ver,onecanapproximare
rhematrixoH (r)o(r) by thediagonalmatrix (rl.t
l",l') t"
to avoid the computationof the inverseinvolved in fI. This approximationis well justifled sincethe off-diagonaltermsof thematrix DH (r)D(r) arenegligiblecomparedto its
diagonalelements(seethe Appendix for further details).Using this assumption,the term
by :
;ltrace (nfr"\- / is approximated
nantcomparedto theterm Df:rtn (3)
o'
\
L,rru""(nfrr)
\
/
", lLi:o'lr,l,)
t u",(o'1r7fi."o1r7)
.
\
(s.48)
Lastly, consideringall theseobservations,an approximationof the actualcompressedlitermsdiscarded,is givenby :
kelihoodfunction,21.;, witn unnecessary
TlrS: \rrur" (o1r1o"@fr.")
.tMNP
\MNo2 L
k:l
\L
p:I
j%r(t - 7)ro
*",,"*o{
\ "w,nl
P
: \ - . 1- ,1 r o ) ,
,)
(s.4e)
P:I
100
where
-i2n(t-r)r}
"rr,rl,
*"-,""0 {
MN
I(r):
\-
./--r
MNo2
(s.50)
can be evaluatedusing the Fast Fourier Transform (FFT). Hence, the normalizedIF is
selectedasfollows :
flf:' ""n {prl(r)}
g'rr(r) :
(.Le*p{ pr l( r ) } dr ) "'
(s.s
1)
which is the product of P elementaryfunctions,each of which dependingon the delay
of a given single path. Here, we succeedin making the different delays separableand
distributedaccordingto the samepdf p(.) givenby :
y\t)-
exp {p11(r)}
I t e x p{ p 1 I( r ) } d r '
(s.s2)
Hence,thejoint pdf of the delaysin g'rr(.) is sptit into the productof P individualpdfs,
which transposesthe problem of generatinga P-dimensionalrandom variableto the generation of P one-dimensionalrandom variablesaccordingto a simpler common distribution.Moreover,the constantterm p1 in (5.51)and (5.52)is differentfrom p6 sinceit is
more advantageous
to use two different valuesas explainedlater. We mention here that
the estimationperformancedependson thesetwo parameters.Actually, the pdf p(.) in
(5.52) exhibits P lobes centeredat the location of the true time delay.But the estimation
enor tn makesother undesiredlobes appearwhich in turn biasesthe generatedvalues
not faithful (spuriousvalues)to the true delays.For this reason,p1 is increasedto make
the pdf p(.) more peakedaroundthe actualdelays{"r}F:, so that the undesiredlobes disappear.However,very large valuesof p1 may also destroysomeuseful lobes and hence
their correspondingdelays will not be generated.Therefore,the optimal value of p1 is
the highestone for which the pdf p(.) still exhibitsat leastP main lobes.Moreover,one
shouldkeepin mind that the normalizedIF in (5.51)is built upon an approximationof the
actualcompressedlikelihood function, which we aim to maximize. Consequently,a bias
will always appearin the mean of the valuesgeneratedaccordingto the normalizedIF.
Fortunately,this biasis alleviatedby the weightingfactorLL,or(.)I g'(.) introducedby the
conceptof IS. Therefore,we maximize the contributionof the compressedlikelihood in
the weighting factor ratherthan its approximationby making p6 higher thanp1. Thus, an
appropriatechoiceof thesetwo parametersreducesthe number,fi, of requiredrealizations
and ultimately the computationalcomplexity.
To summarize,thenew IS-basedML estimatoris givenby :
:
,r:
R
t1 \-
r^^r
rl
-r,po(T*)
'
E L"k\p) g,eJr i
101
(s.s3)
wherez6(p) is thepth elementof the vectorza. We alsomentionthat in practicethe delays
are confinedin the interval 10,LT.l [6]. This allows us to further use the circular mean
insteadof the linear mean givenby (5.53),as detailedbelow.
In few words, to definethe conceptof the circular mean,considera random variable X
taking valuesin [0, 1] accordingto a given distributionG(.). The circularmeanof X is
givenby [24] :
E"{x}: }z
lo'","*'Gt)dr,
(s.s4)
wherethe operatorl(.) returnsthe argumentof any complexnumber.Then, if we havea
set of R realizatio,nS,
/1, . . ., r R, drawnaccordingto the pdf G(.), the circularmeanin
(5.54)is computedas :
11
E-{
" ' t XI
" ) :
R
i-lj-f
2^- R.L
k:1
"r2r'rt.
(5.ss)
Adopting the conceptof circular meanin our problem, an alternativeformulation of the
estimatein (5.53)is givenby :
?o:*t+* Ftr).*o{*#\,
(5.56)
where the delaysare transposedto the interval [0, 1] after being normalizedby LT. and
F'(.) is the weightingfactordefinedas :
(5.57)
Note that the estimatorin (5.56)relies on finding the anglesof a complexnumber.Therefore, we no longerneedto computethe two positivereal normalizationfactorsI,
I, exp{pL.(r)}
and (f exp{p1I (r)} ar)P sincethey canbe droppedwithout ultimatelyaffectingthe final result.Moreover,during the computationof the weighting factorF(.), the exponential
terms in the numeratorand the denominatorof F (.) may result in an overflow.To avoid
this overflow,we substituteF'(.) bV F'(.) givenby :
- ,,*I?x(p))-,?,?l(0,r.?,)- ,,f.Lf',tr)))
F'('r)- exp
}t.tt,
{o,rJ,n
by multiplying F(.) by a positive number.In such a way, the exponentialargumentin
(5.58)no longerexceedszero,alleviatingtherebyany computationdifficulty.
Summary of steps
In the following, we summarizethe entire stepsof the new IS-basedML time delay estimator.
102
dr
1. From the samplesmatrix, Z, computethe periodogram1(.) expressed
in (5.50)at
discretepoints of the interval 10,LT.]. Then evaluatethe elementarypdf asfollows :
exp(p11(1))
It\tzl
-
Df:rexp(p1l (r1,))'
where we substitutethe integral in the denominatorby a summationover all the
discretepoints in the integrationinterval.
2. Generateonerealizationof the vectorr accordingto the pseudo-pdfg'pr(.).Tosimplify, we exploit thefact that the delaysareseparableing'or(.)andwe generateP realizations{rr(p)}'o=, accordingto p(.) usingthe inverseprobabilityintegration[20].
It is importantto makesurethat the P generatedentriesr*(7), Tk(2),. . . , rn(P)
(in order to obtain one vectorrealization)are different.This condition is necessary
sincethe delaysof the pathsarein practicedifferenttherebyensuringthat the matrix
inverse(DH G)D(r))-t in L"(.) alwaysexists.
3 . Repeatstep 2) R - I times then evaluatethe weighting factors F'(tl) for i, :
1,..., R.
4. Find the maximum likelihood estimate of the delavs usins the circular mean tn
(5.56).
5.6 ExtensionTo MC-CDMA Svstems
In MC-CDMA transmitter,the original data are spreadover different subcarriersusing
a spreadingcode. Therefore,it is possibleto transmit severalDS-CDMA waveformsin
parallel.At time indexn, the inputinformationis first convertedinto l/" : 2K 11 parallel
sequencesand modulatedat rate 7lTuc, whereTuc : l/"7 is the symbol durationafter
seriaVparallelconversion.Each of the parallel streamis then spreadedwith a spreading
code at rute Lf T" and modulatedby the inversediscreteFouriertransform(IDFT).
At thereceiver,a reformulationof thepost-correlationmodelfor MC-CDMA of the spatiotemporalobservationfor the kth subcarrierand thenth observationis givenby 126l:
Zk,n: s1,,nJp,nD[(z)
+ l/r,,,
(5.60)
wheres7r,,and Jp,n arethe signalcomponentand the spatialresponsematrix onthe kth
subcarrier,
respectively.
Thecolumnof thetimeresponse
matrixDn(r) : [dr,r, dn,z,. .. , dr,p]
are given by :
dk,p:
"-iark\ffi|p"?ro),
p.(T"lk" - ro)ei2"k&,..., p"((Lk" - I)T"Ik" - ro)"j'"*x#!1t,
(s.61)
103
where) determinesthe frequencyspacingbetweentwo adjacentsubcarriero(f n : ^k ITMC)
andk" is the oversamplingratio [26]. Note herethat thepropagationtime-delaysaresupposedto be the samefor all subcarriers.Unlike the single-carriercase,the receivedsamples
Zp,n ca;rrrrot
be directly used as an input to the algorithms.Therefore,we introducethe
intermediatetransformationof the samplesmatrix, denotedZf,.,, givenby :
ZE,n:Zn,* (" tTo)
: sk,nJk,,D{ (r) + N;,*,
whereLM :
[1,...,l]t anda :
Dft(r) is:
fr, e-i2"+ ,. . . , "-i2"\#E!]".
(5.62)
The prh columnof
dtk,o:d1r,ox a'
-
p.(T"lk"- ,r),
"-tznk\# lp.?ro),
, p.((Lk"- t)T.lk"- r)lr.6.AZ7
Hencewe eliminatein Dft(r) the dependence
row-wiseof thephaseslopeof eachcolumn
vectoron k in the spectraldomain.While the formulationin (5.62)seemsto be adaptedfor
the estimationprocess,the phaseshifte-i2"k^ffi oneachcolumnof Dft(r) set against
the direct use of the formulation in (5.62). To overcomethis problem, we note that the
matrix Dfr(r) canbe writtenas Df(r) : AnD(r) where,46 is a diagonalmatrix which
Then we insert the phaseshift in the spatial{"-i'^u^#\::,
responsematrix Jp,,to obtaina formulationsimilarto the one in (5.3).Finally,to exploit
diagonalelements*"
the frequencygain, we gatherthe transformedobservationover the different subcarrier
into the following compactrepresentation
:
z::lziT, z;,7",., 2fl,,1
: D(r)Jf; + N{ ,
(5.64)
[Ar4r, ArJTr,..., A9.JK.,,,] and N{ : [.Aff;, Nfl,... , l/,(f,,,].
The interestingthing with the formulation in (5.64) is that it increasesthe numberof observationsproportionallyto the numberof subcarrierusedin the system.ConsideringN
whereJf,' :
transmittedsymbols,we formulate a compactrepresentationsimilar to (5.4) by concatenatingthe ,A/observedsymbols:
Z :
:
lzi,2i,..., Z.*l
D(r)J{
+ l,r"?,
(s.6s)
6When.\ is equal to 1, the transiverbelongsto the classof multitone (MT)-CDMA, and if it is equal to
I, the transiverbelongsto the family of MC-DS-CDMA.
ro4
whereJ"r : lJf',..., Jff)and 1y'"7: [lIr"t,. . ., .Af,ifl].
Comparedto a schemathat estimatesthe delaysover eachsubcarrierseparately,the proposedmodelpresentsbetterperformance.In addition,we are ableto derivethe corresponding CRLB following the samestepsaspresentedin SectionII. This revealsthat the CRLB
hasa similar expressionas in (5.20)by multiplyingby a factorof lf N..If we denoteby
CRLBo the CRLB when M : .Ay': N" : 1, the resulting CRLB for the MC-CDMA is
givenby :
:
CRLB
ffirRLBo
(s.66)
As a consequence,
the time delay estimationmergesspace,time and frequency.
To obtain an expressionof the CRLB by using the channelcoefficientsmatrix to estimate
the delays,somemodilicationsto the model areneededand developedin Appendix III. In
this case,we seethat the resultingCRLB dependson both time andfrequencycorrelation.
5.7 Simulation Results
In this section,we comparethe performanceof the two proposedmaximumlikelihood estimatorsagainstthe popularRoot-MUSIC algorithmand the CRLB. In all the simulations,
we considera multipath propagationenvironmentwith 3 propagationspaths and we simulatea challenging
of closely-spaced
scenario
delaysequalto 0.L27,0.15?and0.187.
The meansquareerror (MSE) - usedasperformancemeasure- of the threeestimators
is comparedto the CRLB. First, recall that the EM algorithm is iterativein nature; hence
initialization is a critical issue.Therefore,the initial valuesfor this estimatorare selected
as random variables,centeredat the real time delay and with a varianceof 0.057. The
processinggain is fixed at L : 64 and the optimal valuesof the parametersp6 and pl for
the IS-basedtechniqueareequalto 20 and 10,respectively.We alsoassumethat the power
is equallydistributedbetweenthe threepathson average.
First, we considera singlecarriertransceiverwith lll : 4 antennabranchesat the receiver
and one receivedsamples(N : 1) and we comparethe MSE of the two proposedML algorithmsto thoseof the Root-MUSICin Fig. 5.1.We alsoplot theperformance
of theEM
ML whentheinitial valueshavea varianceof 0.187 reflectinglessaccurateinitializations.
Clearly,both the IS-basedand the EM algorithms,for good initializations,outperformthe
Root-MUSIC techniqueover a large range of the SNR. While the two ML algorithms
presentalmostthe sameperformance,
it is suggested
in practiceto use,in this configuration, the EM ML approachsinceit offers lesscomputationalcomplexity comparedto the
IS-basedalgorithm.Indeed,the EM ML estimatorhasthe advantageof performing P parallel maximizations.Therefore,asthe numberof pathsP increases,thereis no additional
noticeablecomputationaltime cost. On the other side, IS-basedalgorithm can guarantee
robustnessto the initial estimates,contrarily to the EM ML.
r05
MSE ofthe 2nd path
MSE ofthe lst Dath
5101520
d, [dB]
MSE ofthe 3rd path
mean MSE ofthe thrce paths
51015
",
[dB]
FtcuRB 5.1 - Estimationperfonnanceof the IS-based,the EM-basedand the RootMUSIC algorithmsfor closely-separated
delays,M : 4.
So far, all the methodsexhibit good performance,with remarkableimprovementsfor
the two new ML estimators.However,a quick study of the EM algorithm and the RootMUSIC revealsthat they arebasedon an estimateof the covariancematrix of the received
signal from the columns of the matrix Z, and the accuracyof this estimatedependson
the number of antennas(which plays the role of the number of samples).Therefore,we
simulatethe performanceof all thesealgorithmsconsideringonly one receiving antenna
elementand keep the other simulationconditionsthe sameas in Fig. 5.1. The results
are shown in Fig. 5.2. ClearIy,Root-MUSIC estimatoris very sensitiveto the number
of receiving antennabranches.Its performancedegradesconsiderablycomparedto the
previouscase.It fails completelyin estimatingthe delayswhich, indeed,is dueto the poor
estimateof Rs. On the other hand,the EM and the IS-basedalgorithmsareless affected,
in this challengingscenario.They still providegoodestimatesregardlessof thechallenging
Fig. 5.2 actuallysuggests
that the ISoperatingconditionsbasedon shortdatasnapshots.
basedalgorithmis evenmorerobustthanthe EM ML estimatorin this configuration.
To further investigatethis issue,we fix the SNR value at 10 dB and vary the number
of antennabranchesM from 1 to 8 with N : 1. The MSE of the three algorithmsversus
M is plotted in Fig. 5.3. As expected,the IS-basedmethodsattainthe CRLB starting
from a small value of ,44contrarily to the Root-MUSIC algorithm and the EM ML. This
meansthat the IS-basedestimatoris well gearedtowardsituationsof reducedantennaarray
t06
MSE ofthe 2nd path
MSE ofthc lst path
I
[dB]
mean MSE ofthe three paths
MSE of the 3rd path
10
1o
5101520
r [dB]
10 -10
0
r
10
[dBj
FIcURB 5.2 - Estimationperforrnanceof the IS-based,the EM-basedand the RootMUSIC for closely-separated
delays,AI - 7.
sizes.On the otherhand,we plot in Fig. 5.4 the MSE versusN, consideringone antenna
branchesin the receiver.Note herethat to seethe effect of the time channelvariation,we
usethe channelestimatematrix insteadof the receivedsignalto estimatethe delays.More
details about the formulation used in the estimationprocessare presentedin Appendix
2. Clearly, the performanceof the three estimatorsis the same starting from N : 5.
But it is not suitableto increaseN sincethe valuesof the delaysmay changefrom one
symbol to another.Usually, a tracking technique(as the one developedin [26]) is used
to continue estimatingthe delays that is why we prefer to use small number of N in
the estimation.In the region of small numbersof snapshots(l/ is less than 5), the MLbasedmethodsperform betterthan the RooTMUSIC algorithmand the gapbetweenthese
methodsincreases
as N decreases.
To evaluatethe impact of the frequencygain in the multicarrier systems,we plot in
Fig. 5.5 the MSE versusthe numberof subcarriersl/". We fix A'I at I and ,A/at 1 to better
illustratethe influenceof the numberof subcarriersN" on the estimationperformance.As
Iy'" increases,the estimationperformanceimprovesto saturatefor high value of -n/".This
salutationcan be explainedby the increaseof inter-carrierinterferencewith ,|y'",due to
the loss of orthogonalitybetweensubcarrierin a multipath environment.We shouldalso
mentionthe similaritybetweenFig. 5.3 - Fig. 5.5 which proovethat the threedimensions
time, spaceand frequencyhave the sameimpact on the estimationperformanceof the
algorithms.This conclusionis further verifiedby the expressionof the CRLB in (5.66)
r07
1456
Number of antcnnr branches
FrcuRs5.3 - MSE vs. number of antennabranchesM for ly' :
subcarrier,
andSNR: 1 0d B .
1 symbol, K :
7
4567
Numbcr ofrcceived sampl€s N
Ftcunr 5.4 - MSE vs. numberof receivedsymbolsl/ for M : 7 antennabranch,K : 7
subcarrier,andSNR: 10 dB.
which is inverselyproportionalto the numberof receivingantenna,the numberof symbols
andthe numberof subcarriers.
5.8 Conclusion
In this paper,we developedtwo implementationsof the ML criterion for the estimation
of time delay both SC and MC air interfacein multipath environment.We distinguishtwo
estimationsprocess: eitherusingdirectly the receivedsignalor usingthe channelestimate
matrix. In the first option we showanalytically,throughthe CRLB, andby simulationsthat
108
FtcuRB 5.5 - MSE vs. numberof subcarrierwith M : 7 andN : 1.
the three dimensions: time, frequencyand spacehave the sameeffect of the estimation
performance.Whereasbasingon the channelestimatematrix, we exploit the time andfrequencycorrelation.While the two proposedmethodsare an implementationof the same
criterion,eachone hasits attractiveadvantages.
We alsoderiveda closed-formexpression
for the corresponding
CRLB in the contextof DS-CDMA. The first estimatorrelies on
the iterativeEM algorithm with a moderatecomputationcost comparedto the grid search
techniquesinceit transformsthe problem of a multidimensionalsearchinto parallel easy
searchesover one-dimensionalspaces.Comparedto other eigen-basedmethodssuch as
the Root-MUSIC, the EM approachexhibits better performancewith a relatively good
initialization, which is an importantissuefor this algorithm that affectsits estimationperformance.
The other algorithmis basedon an entirely different approach.It relies on a global maximizationtheoremandthe conceptof importancesamplingto directly find the global maximum. The IS-basedapproachalsoavoidsthe multidimensionalgrid searchby approximating the actual compressedlikelihood function; splitting it thereby into separableonedimensionalfunctionsof the delays.It doesnot requireany initialization andhenceit does
not sufferfrom performancedegradation.Its performanceis almostequalto that of the EM
algorithm(whichrequiresgoodinitialization),but at the expenseof anincreasein the computationalburden.Moreover,only the IS-basedalgorithmproducesaccurateestimatesfor
a small numberof receivingantennabranches.The performanceof the EM approachalso
degradesconsiderablyin the specificcaseof single antenna(SISO conflguration)where
the Root-MUSIC algorithmfails completelyto estimatethe delays.
109
Appendix1
OH(r)O(r) = >i:i
theapproximation
Justificationof
The diagonalandoff-diagonalelementsof OH (r)D(z)
lr,l'Io
arerespectivelygivenby :
L-l
l " , l t *, : t , 2 , . . . ,P ,
| o H1 r ) o 1 r ) l * ; ^ :t
(s.67)
t:0
L-l
r
IDH(r)D(r)l*,n: t
TL,
2tr
r')
lctl-exp
m-l
(r
vm
))
L
{,
2..
Tn
P TN IT
n.
(5.68)
While it is easyto verify that :
(s.6e)
IDH(r)D(r)],n,, < loH (r)o(r))*,,*,
for nt f n, the inequality in (5.69) does not guaranteethat the diagonal elementsare
indeeddominantcomparedto the off-diagonalelements.To that end,we defineF(.) the
ratiobetween(5.68)and (5.69)asfollows :
E'(A-
L
\4tm'nJ
\ --
exp{*ry'-}
DL-o'lcll2
D,":] l",l'
1
(s.70)
where Lr,n;n -
rrn - rn is consideredas a random variable uniformly distributed in
l-LT", LT.l.We plot in Fig. 5.6 the probabilityof havingF(A,r^,,) ) r for r e [0, 1]
and we verify that the diagonalelementsof OH (r)O(r) are dominant,with very high
probability,comparedto its off-diagonalelements.This justifies the following approximation:
L_L
oH(r)o(r) = I
lr,l'ro.
(s.71)
t:0
Appendix 2
Model used to estimate from the channel coefficientsmatrix
The developmentpresentedin the main body of the article is basedon the observation
matrix Z. In this case,the columnsof Z are uncorrelatedbecauseof the presenceof
110
ts
n0
F
f:
<0
-5
90
ao
0.1
0.3
0_5
T
o.7
0.8
FrcunB 5.6 - Complementary
cumulativedistributionfunctionof the ratioF(Lr^,,).
uncorrelatedtransmittedsymbols.But if we went to use the channelcoefficientsmatrix
fi : lfrT, fr[,. . . , frT,], thesesymbolsare no longerpresentsand the columnsof
ff ur" correlated.So we introducea smallmodificationon the formulationto remainthe
two algorithmsvalid. This problem can be solvedif viewed in the proper way. First, we
performa column-by-column
FFT of {ff"}I:rto
f(., :
obtain:
D(r)$+t,
:
D (t)l g"( r ) , g"( 2) ,.. ., g"( A' [)+] e*,
(s.72)
where the matrix t,, is the resulting noise and g,(i) is the 'ith column of $ . From the
formulationin (5.72),we bring togetherall the channelcoefficientin onematrix fi. defined,
AS:
(s.73)
g(2),.. ., g(M)l+ t,
fu : D,noaorlg!),
in whichS(i) : l7r(,)', gr(i)',. . . , gN(i)rlT andDnod,if: IxED,where theoperator
I standsfor the Kroneckerproduct.If we considerthat the signalis transmittedthrougha
Rayleighchanneland we denotethe maximum Doppler frequencyby f n, the covariance
matrixof 9(f ) is givenby :
Rs:
(
Jo(0)
Js(21r(N- t)/"7)\
:
:
I
\.r6(2n(N
- t)f oT)
/o(0)
I a E,
(s.74)
)
Taking into account(5.73) and(5.74), the implementationof the two algorithmsusing the
channelinformationmatrix is straishtforward.
t11
Appendix 3
CRLB for MC-CDMA using the channel coefficientsmatrix
We use a similar developmentsas presentedin Appendix II, with the differencethat here
we deal with time and frequencycorrelation.We defineiL"r,, u, the columnby column
FFT of the channelresponsematrix fr;,
fot thekth subcarrierandnth observationwhich
is givenby :
fu'rn :
o(r)Jflt €n,n
:
D ( t ) l g r , " ( I ) , g k , , ( 2 ) , . . , g k , , ( M ) l+ 8 r , " ,
(s.75)
wheregn,"(i) is the i'h column of Jffi. Then we gatherthe channelcoefficientsin the
matrix fL' givenby:
iL":DWG),oe),...,s(A[)],
(s.76)
w i t h g - ( ?: )l s T , r ( i ) , . . . , s T , r . t ( d g) ,T , r ( i ) , . . . ,s T w " , z ( i ) , . .s. T
, , ^ r ( i ) , . . .g, T t " , w ( ia) ]n d
D : Ix 8Iu" I2. Denoteby Q(Lf,Al) the autocorrelation
of the channeltransfer
function (we supposehereuncorrelatedscatteringwherethe autocorrelationtransferfunction in frequencyis a function of only the frequencydifferenceLf [27], the covariance
matrix of g'(d) is Rn. : iD E -R.7wherethe elementsof iD are function of /(A/, Al).
Injecting Rs" andD in6.ZO), the CRLB can be written in this alternativeway :
('))
(cnrn-r
" : d:f."."
f*,,;.ri,ilri*r:'"r,"!,,
_q
o"
L
\\
\
6.77)
whereR71"is the covariance
of ?L" andD; andD;are the derivativeofD and2 with
respectto ri, respectively.If we denoteby Ba the kth P x P block on the diagonalof
nn.Dn4LDRn",weget
(cnI-n-1(z)),- :
'4n{E
@f$-n).,0)
"r(n,i)\,
(s.78)
then we obtain the following expressionof the CRLB :
CRLB(z) :
#1.{f u'0-r)u."-}]-'
tt2
(s.7e)
I
Under certainconditions,iD can be easily expressed.
In fact, d(Lf ,At) is given by
l27l:
dAf.4,) :
I_-J
f ,r. Ltle-i2n^r'dr.
(s.80)
where $.(r, Lt) is the autocorrelationfunction of the channelimpulse response.If the
differentpathshavea Rayleighdistribution,d"(t, Al) canbe writtenas :
Q"(r,Lt! :
J
:
/n+j2rL'fr4,
I_cx r1r7lo(2rf pLt)e-
Js(2rfpLt)er(Lf)
(s.81)
ThenQ : J I Fy whereFr(t, j) : Ft((i - j)AlT*c).
We verify that in the absenceof time and frequencycorrelation,the expressionof the
CRLB obtainedin(5.79) is the sameasin (5.66).
113
Bibliographie
t 1 l K. Cheikhrouhou,S. Affes and P. Mermelstein,"Impact of synchronizationon performance
in widebandCDMA networks",IEEE J. Select.Areasin Comm.
of enhancedarray-receivers
Dec.2001.
vol. 19,no 12,pp.2462-2476,
l2l I. Shakya,F. Ali, andE. Stipidis,"High UserCapacityCollaborativeCDMA," IET Commun.,
vol.5, no.3, pp. 307-3I9,Feb.20l1.
t3l B. Suard,A. Naguib,G. Xu andA. Paulraj,"Performanceof a wirelessMC-CDMA system
with an antennaarray in a fading channel: reverseLink,"IEEE Trans.on commun.,vol. 48,
pp. 1257 -1261,Aug. 2000.
l4l B. H. Khalaj, A. Paulrajand T. Kailath, "2D RAKE receiversfor CDMA cellular systems,"
Proc. GLOBECOM' 94, pp. 400-404,Nov. I 994.
[5] "Performanceanalysisof multicarrier DS-CDMA with antennaarray in a fading channel,"
WirelessPers.Commun.vol. 52,pp.273-287,2010.
'A
t6l S. Affes, P. Mermelstein, new receiverstructurefor asynchronousCDMA : STAR-the
IEEE J. Select.Areas in Comm.,vol. 16, no. 8, pp. 1411spatio-temporal
array-receiver",
1422,Ocl 1998.
UI R. O. Schmidt, "Multiple
emitter location and signal parameterestimation,"IEEE Trans.
AntennasPropag.,vol. 34, no. 3, pp. 276-280,Mar. 1986.
t8l R. Roy and T. Kailath, "ESPRlT-estimationof signal parametersvia rotational invariance
techniques,"
IEEE Trans.Acoust.,Speech,Sign.Process.,vol. 37, no. 7, pp. 984-995,Jul.
1989.
l9l M. Feder and E. Weinstein"Parameterestimationof superimposedsignalsusing the EM
algorithm,"IEEE Trans.Acoust.,Speech,Sign.Process.,vol. 36, pp. 477-489,Apr. 1988.
'A
tl0] M. Pincus, closedform solutionfor certainprogrammingproblems,"Oper Research,pp.
690-694,1962.
111l A. F. M. Smith and A. Gelfand,"Bayesianstatisticswithout tears: A sampling-resampling
framework,"AmenStatist, vol. 46, pp. 84-88,May 1992.
I14
'An exactmaximumlikelihoodnarrowbanddirectionof arrivalestimal12l S. Sahaand S. Kay,
torl' IEEE Trans.Sign.Process.,vol. 56, pp. 1082-1092,Ocl 2008.
'A maximum likelihood angle-dopplerestimatorusing importance
t13l H. Wang and S. Kay,
sampling,"IEEE Trans.Aerospaceand Electro.Sys/.,vol. 46, pp. 610-622,Apr. 2010.
'A
maximumlikelihood
t14l A. Masmoudi,F. Bellili, S. Affes andA. St6phenne, non-data-aided
time delay estimatorusing importancesamplingl'IEEE Trans.Sign.Process.,vol. 59, pp.
4505-4515,
Oct.201l.
'Adaptivespace-time
processing
for wirelessCDMA," in Adapt15l S. Affes andP.Mermelstein,
tive SignalProcessing: Applicationto Real-WorldPromblems,J. BenestyandA.H. Huang,
Eds.,Springer,Berlin,Jan.2003.
tl6l A. D. Dempster,N. M. Laird andD. B. Rubin,"Maximum likelihoodfrom incompletedata
via the EM algorithml'J. Roy.Stat.Soc.,vol. B-39, pp. l-37,1977.
'A new method for resolution estimationof time delay,"Proc.
ll7) Z. Q. Hou and Z. D. Wu,
ICASSn pp. 420-423,1982.
tlSl M. A. Hasan,M. R. Azimi-Sadjadiand G. J. Dobeck,"Separationof multiple time delays
usingnew spectralestimationschemes,"
IEEE Trans.Sign.Process.,vol. 46,no.6, pp. 15801590,June1998.
t19l M. I. Miller and D. R. Fuhrmann"Maximum-likelhoodnarrow-banddirectionfinding and
the EM algorithm," IEEE Trans.Acoust., Speech,Sign. Process.,vol. 38, pp. 1560-1577,
Sept.1990.
t20l S. Kay,Intuitive Probability and RandomProcessesUsing MATLA4 New York : Springer,
2005.
prot21l M. I. Miller and D. L. Snyder,"The role of likelihoodand entropyin incomplete-data
blems : Applications to estimatingpoint-processintensitiesand Toeplitz constrainedcovariances,"Proceedings
of theIEEE, vol.75, pp.892-907,July 1987.
o'Maximumlikelihooddirectionfindingfor stochasticsources: A separable
soI22l A. G. Jaffer,
lution,"in Proc.IEEE Int. Conf.Acoust.,Speech,Sign.Process.,pp. 2893-2896,Apr. 1988,.
t23l W. B. Bishop and P. M. Djuric, "Model order selectionof dampedsinusoidsin noiseby
predictivedensities,"
IEEE Trans.Sign.Process.,vol. 44, pp. 6ll-6l9,Mar. 1996.
124) K. V. Mardia, Statisticsof Directional Data, New York : Academic,1972.
125) P. Stoica and A. Nehorai, "Performancestudy of conditional and unconditionaldirectionof-arrival estimation,"IEEE Trans.on Acoustics,Speechand Sign.Process.,pp. 1783-1795,
Oct. 1990.
115
'A
multicarrierCDMA
t26l B. Smida,S. Affes, L. Jun, and P. Mermelstein, spectrum-efficient
array-receiverwith diversity-basedenhancedtime and frequency synchronizationl'IEEE
Trans.WirelessCommun,vol. 6, no. 6, pp. 2315-232'1,
June.2007.
J. G. Proakis,Digital Communications,4th
ed.McGraw-Hill,2001.
116
Conclusion
Dans ce m6moire,le probldmede synchronisationtemporelleen communicationnum6rique a 6t€ trait6.Nous avonsd6velopp6desproc6duresd'estimationdu d6lai de propagation bas6essur le critdre de maximum du vraisemblancepour diverssystbmesde communication.Le premier estimateurest ddvelopp6pour les signauxmodul6soir les symboles
6mis sontinconnus.La m6thode"importancesampling"(IS) est adaptdepour trouverl'estim6 h maximumde vraisemblance.
Bien qu'il soit d6velopp6sousI'hypothbsed'un seul
trajet de propagation,il trouve des applicationsdansplusieurssystdmestels que les communicationssatellitaires.Une extensionau casmulti-trajetsest aussiproposde.Danscette
nousnousretrouvonsfaced plusieursd6laisd estimer.Bien quela fonction
configuration,
de vraisemblancesoit multidimensionnelle,I'IS offre une proc6dureattirantepour transformer le problbmemultidimensionnelen un probldmeunidimensionnel.Autre contribution, nousavonsd6velopp6deuxalgorithmesde synchronisationpour les systbmesCDMA
mono-porteuse
et multi-porteuses.Le premieralgorithmereprendle principede I'IS au niveaudu signalaprdsd6s6talement.
L'autre algorithmesebasesur la m6thode"expectation
maximization"(EM) qui transformele problbmemultidimensionnelen de simplesop6rations unidimensionnellesdont le nombreaugmentelindairementavecle nombrede trajets
d6tect6s.Bien que les deux m6thodessoientdes impl6mentationsdu mOmecritdre, chacunepossddesesproprespoints forts. La m6thodeEM estimedes d6lais de propagation
avecune complexitdrelativementfaible compar6ed dSautresalgorithmespuisque,tel que
d6montrerdans ce mdmoire,l'estimation des diff6rentsd6laispeut se faire en parallble,
ce qui r6duit le tempsde calcul. Les simulationsmontrentque cet algorithmepr6sentede
queles techniques
meilleurperformance
de sous-espace
tel quele Root-MUSICet ceci en
utilisantde bonneinitialisations,qui doit 0tre soulign6commeun point faible de l'algorithme.La performancede I'algorithmeEM d6gradeconsid6rablement
dansle casd'une
seuleantenner6ceptricealors que I'algorithme Root-MUSIC 6chouecomplbtementd estimer les d6lais.D'un autrecot6,I'algorithmeIS ne n6cessite
aucuneinitialisationet offre
de bonneperformancemOmeen pr6senced'une seuleantenner6ceptrice,et ceci au prix
d'unecomplexit6accrue.Nousmontronsaussipar simulationsqueles dimensionsespace,
tempset fr6quence(pour les systbmesmulti-porteuses)ont le mOmeeffet sur les performancesde cesestimateurs.
I-lautrevolet trait6 dansce rapportestla d6rivationdesexpressionsanalytiquesdesbornes
It7
de Cram6r-Raodes estimateursnon biais6sdu retard pour les modulation BPSK, MSK
et QAM carr6esen estimationaveugle.Ces expressionsanalytiquesr6vblentque les performancesd'estimationdu d6lai ne d6pendentpas de la valeur du parambtreen question,
chosequ'on ne pouvait pas confirmer auparavanten dvaluantles bornesde Cram6r-Rao
par desm6thodesempiriqueset que I'estimationdu ddlaiest independent
de l'estimation
de la phaseet de la fr6quence.On dit que le d6lai est d6coupl6de ces deux derniersparamdtres.Nous avonsaussid6riv6les expressions
de cesbornespour les systbmesSC- et
que les trois dimensions: spatiale,tempoMC-DS-CDMA. Dansce cas,nousconstatons
relle et fr6quentielle,agissentde la mOmefagonsur les performancesd'estimation,se qui
confirmeles r6sultatsobtenuspar simulations.
Cependant,plusieursextensionsrestentd explorer.Dansce m6moire,nousavonstoujours
suppos6que le bruit additif 6taitblanc et Gaussien.Ce seraintdressantde voir les modificationsh faire, que ce soit au niveaudesestimateursde desCRLBs, si le bruit est color6.
De plus, le canalde propagationdansles chapitres1 et 3 est suppos6constant.UadaptaAussi,la CRLB
tion d'un canalvariablerefldtemieux larlalitf du canalde transmission.
est d6velopp6edansle chapitre2 sousI'hypothbsed'un seul trajet de propagation.La
ddrivationde la CRLB dansun canal d multi-trajetsresteun bon sujet de recherche.En
ce qui concernele 5em chapitre,nous avonssuppos6que les antennessont d6corr6l6es.
U6valuationde la robustesse
des estimateursd6velopp6sdansce chapitreen cas dSantennescorr6l6esreste d venir et le d6veloppementd'estimateursqui tiennentcompte de
cettecorr6lationseraitpeu 6tre d'actualit6 si jamais nousremarquonsune d6gradationde
performancedansce cas.
118