Design of simulation platform joigning site specific radio propagation ...

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Design of simulation platform joigning site specific radiopropagation and human mobility for localization

applicationsNicolas Amiot

To cite this version:Nicolas Amiot. Design of simulation platform joigning site specific radio propagation and hu-man mobility for localization applications. Other. Université Rennes 1, 2013. English. <NNT :2013REN1S125>. <tel-00971809>

https://tel.archives-ouvertes.fr/tel-00971809

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ANNÉE 2013

THÈSE / UNIVERSITÉ DE RENNES 1 sous le sceau de l’Université Européenne de Bretagne

pour le grade de

DOCTEUR DE L’UNIVERSITÉ DE RENNES 1

Mention : Traitement du signal et Télécommunications

Ecole doctorale (MATISSE)

présentée par Nicolas AMIOT

Préparée à l’unité de recherche IETR- UMR CNRS 6164 (Institut d’Electronique et de Télécommunication de Rennes)

(groupe Propagation Localisation)

Design of a Simulation

Platform Joining Site

Specific Radio

Propagation and

Human Mobility for

Localization

Applications

Thèse soutenue à Rennes le 2 Décembre 2013

devant le jury composé de :

Laurent CLAVIER Professeur, Télécom Lille / rapporteur Rodolphe VAUZELLE Professeur, Université de Poitiers / rapporteur

Benoît DENIS Ingénieur Docteur, CEA-Leti / examinateur Emanuel RADOI Professeur, Université de Bretagne Occidentale / examinateur

Bernard UGUEN Professeur, Université de Rennes 1 / directeur de thèse Mohamed LAARAIEDH Enseignant Chercheur, SupCom Tunis / co-directeur de thèse

P ss

s♥ ♦ ♠t♦♥ Pt♦r♠ ♦♥♥ t ♣ ♦

Pr♦♣t♦♥ ♥ ♠♥ ♦t② ♦r ♦③t♦♥ ♣♣t♦♥s

♦s

♠r

♦s

s♥ ♦ ♠t♦♥ Pt♦r♠ ♦♥♥ t ♣ ♦ Pr♦♣

t♦♥ ♥ ♠♥ ♦t② ♦r ♦③t♦♥ ♣♣t♦♥s

s P tss s ♥ ♠r t ❯♥rst② ♦ ♥♥s

ssss♠♥t ♦♠♠tt

Pr♦ r♥t ❱ r♥Pr♦ ♦♦♣ ❱❯❩ ❳ P♦trs r♥P ♥ ♥♦ît r♥♦ r♥Pr♦ ♠♥ rst r♥

♣rs♦rs

Pr♦ r♥r ❯❯ ❯ ♥♥s r♥P ♦♠ ♣♦♠ ♥s

♦ ♥ ♠ ♥ ♠② ♣r♥ts

♥♦♠♥ts

rst ♦ ♦ t♦ t♥ ♠② tss s♣rs♦r Pr♦ss♦r r♥r ❯♥ ♦rs ♥♥ ♣t♥ tt♥t♥ss ♥ t② ♦r t st tr ②rs s♦♥ t t♥ ♠ ♦r t ♥♦ ♥ ♦♥ t♦ ♠ ♦r ♠♦st rsts ♣r♦ss♦r ♥ s st ♥ ♥♥r♥ st♥t ♥ t♥ s ♦ ♥ r ♠ s rsr ♥♥r r♦r t♥ ♠ ♥♥t② ♦r ♥ ♥t♥♠② ♦ tr♦♥s ♥ ♣rtr ♥ s♥ ♥ ♥r

♦ s♦ t♦ t♥ ♦♠ r ♠② ♦ ♥ ♦♥t s♣rs♦r♦r s ♣ r♥ t ②rs sr t s♠ ♦ ♦ r② t♦ ♣②trt t♦ s ♣r♦ss♦♥s♠ ♥ r♦r t♥ss r♦♠ t ♥♥♥ ♦ stss ♥t t♦② ♠♦♥ts ♦ ♦♥ ♥ ♦ ♥ s ♠ tst ♦ s♦ t♦ s♣r ♥ t♦t ♦r s ♥ tr ♦ s ♦r♥r♥ ♦r ②rs ♦ ♦♦rt♦♥

♥ t ♦rs ♦ ts tss Pr♦ss♦r r♦s Prs♥ r♦♠ t ❯♥rst② ♦ ♦r♥♠r s ♥t t♦ s♣♥ ♠♦♥t t t ♦rt♦r② ♥ t r♠♦r ♦ tr♦♣♥ P ❲r ♣r♦t ♦ t♦ t♥ ♠ ♥♥t② ♦r t ♣ ♣r♦ t ♥trst s♦ ♥ t t♠ ♦t t♦ ♠ r♥ ts ♣r♦ s♠t♠t r♦r ♦♠♥ t s ♥♦ s ♣② ♦♥trt t♦ tr♥s♦r♠♠② ♦r ♠t♦s ♥ t ② ♥ s♥t sss

♥ t ♦rt♦r② ♦ t♦ t♥ té♣♥ r♦♥ ♦ s ②s ♥tt♥t ♥ ♦ t♦ t♥ ♦ë r srtr② ♦ t ♣rt♠♥t♥ ♣② trt t♦ r ♣r♦ss♦♥s♠ r♦r t s♦ r ♦t♦♥ ♥ ♠♥t② ♦ s♦ t♦ s♣r ♥ t♦t ♦r ♠② tss ♦s t ♦♠ sr♦r sr ♠② ♦ r♠ ♦♥♥ r♥ ♦①♥ P♣ ♦ss ♥ ♦ ♣t ♣ t ♠ ♦r ts ②rs

♥ ♠♦r ♣rs♦♥ ♦ t♦ strt ② t♥♥ ♠② r♥s ♦r tr♠♦r s♣♣♦rt ♥ ♦r t ♦♦ t♠s s♣♥t t♦tr ♦ t♦ strtt t♦s ♦ r ② ♥ ♦ ♥♥♦t s s ♦t♥ s ♦ t♥ ♦sé♦r ♥ ♥ t r♥ ♦r ②rs s s s ♣rt♥r r♦tt ♦r trr♠ ♦♠ ♥ t ♦♦ t♠s sr t♦tr s♦ t♥ ♦ ♥ r♦♥♦ r ♥② ♥♦②♥ ♥ ♦r① ♣♣rt tr s♣♣♦rt ♦♦ ♠♦♦ ♥ t ♥ t♦tr ♦ s♦ t♦ t♥ t♦s ♦ r ♦sr ②♥ ♦ s♦ ♥rtss s ♠♦r ♦t♥ ♥s t♦ r♥ ♥ ♦s Prr ♥❱ér r ♥ ré ♦r t ♠♠♦rs r♦♠ ♥ r t ♥♥r♥ s♦♦♥ t ♣rts ♥ ♥♥s ♦ ♠s ♥ ♥ ♠♥t♦♥

st t ♥♦t st ♦ t♦ t♥ ♠② ♣rt♥r ♠ ♦ ♥♦rs s♣♣♦rts ♥ ♣ts ♣ t ♠ ♥ s ♠ t str♥t ♥ t♦ s ♦r t♥ r ♦r ♥ ♠ t ♦ ♥ tt♥t♦♥ ♥ tt s t s ♠ tstr♥t ♦r ♥ t♦ s ♥ ♦ ♦rs ♦r ♣♣♥ss ♦ ♥♦t ♦♠♣t t♦t ♦r s♦♥ ♥ ♦ s ♥ t♥ ♣ ♦r s ♦r t ♣st ②r ♥ ♦ ♥♦t ♦ t♦t t♠ ② ♠② s ♥② ♦ t♦ t♥ ♠②♣r♥ts ♦r ②s ♥ ♠ tr s♣♣♦rt ♥ ♦

♦♥t♥ts

♦♥t♥ts

♥♦♠♥ts

♦♥t♥ts

♥tr♦t♦♥

tr♠♥st Pr♦♣t♦♥ ♦♥ ❱t♦r③ r♣s ♣♣r♦ ♥tr♦t♦♥ ♦♥t①t r ②r♥ ❯tr ❲ ♥ ❯❲ ♥ Pr♦♣

t♦♥ ♦♥ t Pr♦♣t♦♥ ♥♥ ♥ r ♦ t rr♥t s ♣ts ♦ t ❯tr ❲ ♥ ❯❲ ♥♦♦② ♥rts ♦♥ t Pr♦♣t♦♥ ♥♥ ♣ts ♦ t ❯❲ Pr♦♣t♦♥ ♥♥ tr♠♥st ♦ ♦ t Pr♦♣t♦♥ ♥♥

②♦ts r♣s sr♣t♦♥ r♣s ♦tt♦♥ r♣s ♦r ②♦t sr♣t♦♥ r♣s ♦r ♣♥ tr♠♥t♦♥ ♦ ♥trs

r♦♠ r♣s t♦ ② ♥trs ♦tt♦♥ ♦ t ♥tr♦t♦♥ ♦ ② ♥tr t♥♥ t ♥trs r♦♠ Gi

tt♥ ②s ♦t ♦ ♥trs tt♥ ②s ♦t ♦ ♥trs r♥ ②s r♦♠ ②s

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥ t ♠♥s♦♥ rr② ♦♦s♥ ❱t♦r③ t r♥③t♦♥ ①♣rss♦♥ ♦ s ♥trt♦♥s t♦♥ ♦ ♥trt♦♥s Pr♦♣t♦♥ ♥♥ ♥ r♥s♠ss♦♥ ♥♥

t ♣ t ♦ ①st♥ t ♠♦s

♦♥t♥ts

♦sss st♠t♦♥s ①ss ② st♠t♦♥ t♦♥ ♦ t t♦♥ ♥ r♥s♠ss♦♥ ♦♥ts

♦♥s♦♥

♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s ♥tr♦t♦♥ tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

♣♣t♦♥s ♦ ♦③t♦♥ t♥♥ P♦st♦♥ r♦♠ t ♦ srs ♥r♣r♥t♥ s t♦ ♥ s t♦ ♦rt♠s s ♦♥ r t♦ ♦rt♠s s ♦♥ ♥tr ♥②ss t♦s

sr♣t♦♥ ♦ t ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ P ♦♥str♥ts ♥ ♦♠tr ♦♥str♥ts ♥t♦♥s ♦♠tr ♦rt♠ sr♣t♦♥ t♦♥ ♦ t ♦st ♦♠tr P♦st♦♥♥ ♦rt♠

P tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

sr♣t♦♥ ♦ t Pr♦♠ Pr♦♣♦s s♦♥ ♣ s ❯♥♦rrt ss♥ ♦ P♦r rr♦rs t♦♥ ♦ tr♦♥♦s ♦③t♦♥ s♥ ②♣♦tss st♥ ②♣♦tss st♥ t♦ s ①st♥ t♦s

♦♥s♦♥

②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥♥t ♦t② ♥ t t♦r ♥tr♦t♦♥ tt ♦ t rt ♦♥ ♦t② ♥ t♦r ♠t♦rs

②♣s ♦ ♦t② ♠♦s srt ♥t ♠t♦r t♦r ♠t♦♥ ♥ ♦t②

♠t♦♥ ♦ t ♥ts ♦t② r ♦t② r♣ sr♣t♦♥ ♠ ♦t② tr♥ ♦rs

♠t♦♥ ♦ t ♦ t♦r r♥③t♦♥ t♦r r♣ r♥③t♦♥ t♦rs Prs♦♥ t♦r Gp

♠t♦♥ ♦ t ♦♠♠♥t♦♥ t ♦♠♠♥t♦♥ t Pr♦ss♥

♦♥t♥ts

t ❯♣t ♦♠♠♥t♦♥ strt♦♥

①♠♣ ♦ ❯s ♦ t ②♥♠ Pt♦r♠ s♣②♥ t ♦t② ①♣♦t♥ t ♦t♦♥ ♣♥♥t Pr♠tr P ❯s♥ t

t❲ ♣♣r♦ ♦♠♣t♥ t ♥♥ ♠♣s s♣♦♥s ❯s♥ t ②

r♥ ①♠♣ ♦ ❯s ♦ t ♦♠♠♥t♦♥ t

♦♥s♦♥

t♦♥ ♦ t Pr♦♣♦s t♦s ♦♥ sr t ♥tr♦t♦♥ sr♣t♦♥ ♦ t sr♠♥t ♠♣♥

P ❲ Pr♦t ♣ts ♦ t P sr♠♥t ♠♣♥ ♦♥sr ♦ ♥♦♦s sr♣t♦♥ ♦ t t ♥r♦s r♥ tt ♦s r♦♠ ♥r♦

♦♠♣rs♦♥ t♥ ♠sr ♥ s♠t Ps t♦♥ ♦ t t ♣♣r♦ ♦♥ sr t t♦♥ ♦ t ② r♥ ♣♣r♦ ♦♥ sr t

t♦♥ ♦ ♣r♦♣♦s ♦③t♦♥ ♦rt♠s t♦♥ ♦ t Pr♦r♠♥s ♦ P ♦③t♦♥ s ♦♥ ②♣♦tss tst♥

♦♥s♦♥

♦♥s♦♥ ♦♥s♦♥s Prs♣ts ♥ tr ❲♦r

és♠é ♥ r♥çs ♥tr♦t♦♥ ♦ést♦♥ étr♠♥st ♣r♦♣t♦♥ ❯♥ ♣♣r♦ t♦rsé

sé sr s r♣s tst♦♥ s r♣s ♥s ♦t s r♣s ① s♥trs r②♦♥s s s♥trs ① r②♦♥s ♥ tr♥s♠ss♦♥ ♦è t❲ s♣éq ♦♥s♦♥

♦st♦♥ ♥ ♠ ♥térr s ♣♣r♦s r♦sts t étér♦è♥s sr♣t♦♥ ♦rt♠ ♦st ♦♠tr P♦st♦♥♥ ♦

rt♠ P

♦♥t♥ts

♦st♦♥ étér♦è♥ sé sr tst ②♣♦tèss ♦♥s♦♥

♣t♦r♠ ②♥♠q érr ♣r♦♣t♦♥ ♦♠♠♥t♦♥ t ♠♦té s ♥ts ♥s rés ♦té s ♥ts ♦r♥st♦♥ rés ♥ r♣ é♥s♠ ♦♠♠♥t♦♥ ♥tr s ♥ts ①♠♣ tst♦♥ s♠tr ②♥♠q ♦♥s♦♥

t♦♥ s ♠ét♦s ♣r♦♣♦sés à ♦♥♥és ♠srés ①♣♦tt♦♥ s ♦♥♥és ♠sr ❲ t♦♥ s ♣r♦r♠♥s ♦rt♠ P t♦♥ ♦rt♠ ♦st♦♥ tst ②♣♦tèss ♦♥s♦♥

♦♥s♦♥ é♥ér ♦♥s♦♥s ①s rr tr

t ♠♥s♦♥ rr② ♥r ♥♦tt♦♥ ♦♥♥t♦♥ ♣rt♦♥ rs ♣ ♦tt♦♥s ♥ ♣rt♦rs

r♦st♥ ♦♥t♥t ❯♥♦♥ s♣ t♦♥ t♥ trs ♥ s ♦r ss ♦t Pr♦t

st ♦ rs

st ♦ s

st ♦ r♦♥②♠s

st ♦ t Pt♦♥s ss♦t t ts ❲♦r

♦r♣②

♣tr

♥tr♦t♦♥

rss ♦r s ♥r ♥ ♥ s♦ ♠♥② ♥s ♠♦♥ t ♠♦st♠♣♦rt♥t s t♦ ♥ t st s♦t♦♥s t♦ ♦♥t t t♦ ♥t♦♥st ♦♥str♥ts ♦♥ t ①♣♦♥♥t r♦t ♦ t tr♦♣t ♠♥ ♥ t ♥sst② ♦♥ ♦s t♦ r t ss♦t ♥r② ♦♥s♠♣t♦♥ ♦r rss♥ t♦s ♥srss ♥t♦rs r t♦ ♦♠ ♠♦r ♣t ♠♦r s r ♦ ♥r♦♥♠♥t ♥ ♦ s♣tr rs♦rs ♠♦r ♦♣t♠③ ♠♦r s r♥♥ ♣ ♥ ♦r ♠♦r♦♥t ♥ tt tr♥ ♥ t rss ♦♥t①t ② ♥♥ trs tt♣r♦t♦♥ r t ♦③t♦♥ ♥ ♣♦st♦♥♥ ♥t♦♥s

s ♥♦r♠t♦♥ ♦ ♣♦st♦♥ s ♦r♥② ♥ sst ♦r t ♥ sr s ♦tt♦ ♦♠ ♥ sst ♦r t ♦ ♦♣rt♦r s♥ t ♦t♦♥ ♥♦r♠t♦♥ ♦t ♦r♦♥t♥♥ ♦r♥ srs ♦r t ♥ sr t s♣② ♦r ♦♣t♠③♥ts ♥t♦r ♥ tr♠ ♦ tr♦♣t ♥ ♥r② ♦r ♥st♥ t ♥♦ ♦ t♣♦st♦♥ ♦ t sr s ♦s t♦ ♦s t ♥r② ♦ t ♦♠♠♥t♦♥ t♦ trr ♠♦r♠♥ ♦r t♦ ♥t② strt t s♣tr♠ rs♦r t♥ tr♥t s

♥t s ♦ t ♣♦st♦♥ ♥ t ♦♠♠♥t♦♥ s ♥ st ♥r♦♣♥ ♦♦♣rt ♣r♦t s ♦r ①♠♣ t P ❲ ♥ P❲ r♦♠ ts sts ♥ r♦♠ ♠♥② ♦trs t ♣♣rs ttt ♣♦ssts ♦r ② t ♦③t♦♥ ♥ qst♦♥ t ♥t ♥tr③ ♥t♦r♦r♥③t♦♥ t ♦r♥③t♦♥ ①st♥ r♦♠ t ♥♥♥ ♦ ♠♦ ♥t♦r t♦ t♥♣♥t ♥t♦r ♦ ♠♦ t♦ strt ♦r♥③t♦♥ ♥ strt♥t♦rs r ♠♦tt ♦t ② t ♦③t♦♥ ♣ts ♥ ② t ♠t♣t♦♥♦ t ♥♠r ♦ st♥rs ♦♥ s♥

Prs♠♥ tt t tr r♦ ♥t♦rs ♦t strt ♥ tr♦♥♦s♥ tt ♦③t♦♥ ② ♦♥ ts ♦t♦♥s t s ♠♥t♦r② t♦ ♣tt ①st♥ ♦rt♠s t♦ ♦♠♣♥t t t ♣♦♠♥ ♣r♦rss ♦ t s♠t♦♥qst♦♥ rss

♥ ♦ t ♥ ♥♥r ② t♦s ♦t♦♥s s t♦ ♣r♦ s♠t♦♥ t♦♦s t♦ t s ♦♠♣① s♥r♦s s ♥ rqrs ♠t♣ ♠♥tsrst ss♠♥ ♥ ♥rs ♦ ♦t rq♥② ♥ ♥t ♦ t st♥rs s♠

♣tr ♥tr♦t♦♥

t♦r s♦ t♦ s♠t tr r ♥ t♥♦♦s t♦ ♦♠♣♥t t♠♦st ♦ t ♣♦♠♥ ♦t♦♥s ♦♥ t rss ♣t② s ♥r② ♥t♦ ♠♦ r♦ s ♥ tr♠s ♦ s♠t♦♥ t ♠♥s tt t ♠♦t② s t♦ t♥ ♥t♦ ♦♥t ♦r r♦♠ t s♠t♦♥ ♣♦♥t ♦ t ♠♦t② trst♦ ♥s rt♥ rst ♥t ♠♦t② t♦ t♦ ♣r♦ ♦rrt ♦srs ♥trt♥ t s♠t♦♥ ♠♦t② s ♣r♠tr ♦ t ♦ s♠t♦♥r ♥ strt ♥t♦r t ♦♥st♥t qt② ♦ sr s ss♥t② ♥sr ②t ♣t② ♦ t ♥♦ t♦ ♦♠♠♥t t ♦tr ts ♦♠♠♥t♦♥ s ♦r♥rst♦♥ ♠♥s♠s t♦ r③ t r st ♥ ♦♣♥ qst♦♥ r♦♠ ts ♦srt♦♥ t s rqr r♦♠ t s♠t♦r t♦ ♦♥sr ♣♦ss ♦♣t♦♥s ♦r ② s♥t♦rs

♠♥ ♣r♣♦s ♦ ts tss s t♦ ♣rs♥t ♥r ♣t♦r♠ ♠s t♦rss t♦s qst♦♥s r♥♥ r♦♠ t ss ♦ ♣②ss ♦ ♣r♦♣t♦♥ t♦r t♦r ♥②ss ♦ ♦♦♣rt ♥ts ♥ ♥ rst ♥♦♦r ♥r♦♥♠♥t ♦♥tr♦ ♣r♦♣♦s ♦③t♦♥ ♦rt♠ s ♦♥ ♥tr ♥②ss ♠t♦s stss s ♥ tt♠♣t t♦ rs t♥ ♣r♦♣t♦♥ ♥ ♠♦t② ♠♦t② ♥♦rt♠s ♦rt♠s ♥ ♣r♦♣t♦♥ ♥ t ♣r♣♦s ♦ ♣r♦♣♦s♥ ♥ t♦♦s s♥t t ♦③t♦♥ ♥t♦♥ ♥ ♠♥ t ♦ st ♦r r ss ♥tr♦♥♦s ♥t♦r ♦♥t①t

s ♦r s str♦♥② ♥ r♥ ② t ♦♠♠t♠♥t ♦ ♦♦♣rt ♣r♦ts trst t P ❲ ♣r♦t t♥ t P ❲ ♣r♦t ♥ ♠♦r r♥t② t ♣r♦t ♦♥ tr♠ ♦ ♦rts r② t s♦rt t♠ s♦ tss ♥ r ♣rt ♦ t ♦r ♦rr♦ ♠ t♦ ♣r♥t ♦r ♦♥ t s♠

♦♠♥t r

s ♦r srs s♠t♦♥ ♣t♦r♠ t tr st♥t ②rs sts♣♥♥ s♠t♦♥ ♥♦♦r ♦③t♦♥ ♥ ♠t♥t s♠t♦♥ ♦♠♥t♦r♥③t♦♥ rts t ♣t♦r♠ strtr ♥ s ♥t♦ ♦r ♠♥ ♣trs♣trs ♥ rss s♣ ♣♦♥t ♦ t ♣t♦r♠ ♣r♦♣t♦♥ ♦③t♦♥ ♠♦t② ♦♥ ♦ ts ♣trs s s ♦♥sst♥t ♥ ♣r♦s tstt ♦ t rt ♣rs♥tt♦♥ ♦ s♣ ♦r♠s♠ ♥ ♥ t♦♥ ♦ t sr ♠t♦ ♣tr s ♥ tt♠♣t t♦ strts t ♣♦t♥tt② ♦ t t♦♦s♥tr♦ ♥ ♣rs♥t ♦r t ♣r♦s strt♦♥s ♥ ♦♠♣rs♦♥ ♦♥ r♦ss♥r♦s tr ♣r② s ♦♥ s♠t♦♥ ♦r s♦ t♥ ♥t ♦ ♠sr♠♥tst

♣tr s t t♦ t ♣rs♥tt♦♥ ♦ ♥ ♥♦♦r st s♣ r♦ ♣r♦♣t♦♥ s♠t♦r s ♦♥ r♣ r② tr♥ tr s♦rt stt ♦ t rt ♦♥ tr♥t ♠t♦s s t♦ ♠♦ t ♣r♦♣t♦♥ ♥♥ ♥ st♦r② ♥ ♥ ♦r♦ t ①st♥ t♦♦s s ♣r♦ tr ♦♥sr♥ tt ♥♦♥ ♣r①st♥ t♦♦s r♦♥♥♥t ♥♦ t♦ ♦rrt② rss t ♣r♦♠ ♦ t ②♥♠ ♦③t♦♥ ♥ t❯❲ ♦♥t①t r r♠♥r ♦♥ t ♣r♦♣t♦♥ ♥♥ ♣r♦♣rts s ♣r♦♣♦s

♥ t sr♣t♦♥ ♦ t ♣r♦♣♦s t♦♦ strt s sr♣t♦♥ rst ♣r♦♣♦ss ♥♦r♥ r♣ sr♣t♦♥ ♦ t ♥r♦♥♠♥t r♦♠ tt r♣ sr♣t♦♥ r②s rt s♥ ♠♦♥ ♦tr t ♦♥♣t ♦ s♥tr s r② ♥♥♦t ♥ ♦♥♥♥t ♦r ♥r♠♥t ♥ t r②s ♥ ♦t♥ s♦♥ ♣rt ♦ss♦♥ t t♦♥ ♦ t ♥♥ r♦♠ t♦s r②s s ♣rt ♥tr♦s ♥ t♦r③ ♣♣r♦ s♥ t ♠♥s♦♥ rr② s rqr t♦ ♥tr♦ s♣ ♥♦tt♦♥ ♣r♦♣t♦♥ ♥ tr♥s♠ss♦♥ ♥♥ r ts ①♣rss s♥ts ♥♦tt♦♥ ♥② t♥ ♥t ♦ t t♦r③ ♣♣r♦ t ♣r♦ss ♦ sts♣ ♠t ♠♦ s sr s t♦♥ ♦♠♣rs♦♥ ♦ t ♥trt♦♥♦♥ts t♦♥ s ♣r♦r♠ t ♣r①st♥ ♣♣r♦

♣tr s ♦s ♦♥ t ♥♦♦r tr♦♥♦s ♦③t♦♥ ♦rt♠s tstrts t ♥ ♦r ♦ t ♦t ♦t♦♥ strts ♥ rss ♥t♦rs ♥ t ♦srs ♥ t♦ t②♣s ♦ ♦rt♠s r t r s ♦rt♠s ♥ ♦♠tr s ♦rt♠s tr s♦rt ♥tr♦t♦♥ ♠♥t♦♥♥ t♥ts ♦ s♥ ♦♠tr s ♦rt♠ t ♦st ♦♠tr P♦st♦♥♥♦rt♠ P s ♣r♦♣♦s r♥t st♣s ♦ ts ♦rt♠ r t ♥tt ♣rt ♥ t ♠t♦ s t s♥ ♦♥t r♦ s♠t♦♥s ♥ ♦♠♣r t♦ ①♠♠ ♦♦ ♥ ❲t st qr ❲ ♣♣r♦s❯s♥ t ♣r♦♣rt② ♦ t ♦♠tr ♣r♦ss♥ ♦ ♣r♦♥ ♠t♠♦ s♦t♦♥s ♥ tr♦♥♦s ♦t♦♥ ♠t♦ s ♦♥ ②♣♦tss tst♥ s t♥ ♥tr♦ ♠t♦ ♣s t♦ rs♦ t ♠t② ♣♣r♥ ♥ t ♥♠r ♦ r♦ ♦srss ♥s♥t t♦ ♦t♥ ♥q ♣♦st♦♥ st♠t♦♥ ② s♥ t ♣♦r ♥♦r♠t♦♥t♦ t s♦♥ ♠t♦ s ♥② t t ♦♥t r♦ s♠t♦♥s ♥t ♥♦♥ ♠ss ♣ss♥ ♣r♦♠

♣tr rsss sr ♣r♦♠s ♦ t ②♥♠ ♣t♦r♠ ♥♥ t♠♦t② t ♥t♦r ♦r♥③t♦♥ ♥ t ♥tr♥♦s ♦♠♠♥t♦♥ ♣trstrts t stt ♦ t rt ♦♥ ♠♦t② ♠♦s ♥ ②♥♠ r♦ s♠t♦rs ♥ s♠t♦♥ ♦ ♥t ♠♦t② t s s sr t r s t ♥t ♠♦t②s♠t♦♥ ss t r♣ sr♣t♦♥ ♦r ♠♥♥ t st♥t♦♥ ♦ ♥ts ♥t♦ ♥♥r♦♥♠♥t t s♠ s t ♠♦♠♥t ♦ t ♥ts t♦ tr st♥t♦♥s s ♠♥ ② rt ♦r ♠t♦ s ♦♥ t str♥ ♦rs ♥ ts ♠♦t② ssr t ♣tr ♦ss ♦♥ t ♥t♦r rt♦♥s♣s t♥ t ♥ts ♦t② t ♣r♦s② ♠♥t♦♥ ♠t ♣♣r♦ s s t♦ t t ♦t♦♥♣♥♥t Pr♠trs Ps s♠t t♥ t ♥ts ♥ s ♦♠♠♥t♦♥ ♣r♦t♦♦ t♥ t ♥ts ♦ t ②♥♠ ♣t♦r♠ s ♥tr♦ ♥②t st st♦♥ s t t♦ t s ♦ t ②♥♠ ♣t♦r♠ t♦ ♣r♦ rst s♠t♦♥s rst ②♥♠ s♥r♦ ♥♦♥ s♥ ♥t s r♥ ♥ Ps♥ ♥♥ ♠♣s s♣♦♥s r ♣r♦ s♥ t st s♣ ♠t♥ t t♦♦ s♦♥ ②♥♠ s♥r♦ s r♥ t ♥ts ♠♦♥ ♥t♦ s②♥tt ♥r♦♥♠♥t ♦s s♠t♦♥s ♠♦♥strt t ♠♣♦rt③♥ ♦ t ♠♣♠♥t♦♠♠♥t♦♥ ♣r♦t♦♦ t♦ ♦♦♣rt ♦③t♦♥

♣tr s ♦s ♦♥ t ①♣♦tt♦♥ ♦ t P ❲ ♠sr♠♥t♠♣♥ t♦ t t ♣r♦♣♦s ♦rt♠s rst t ♠sr♠♥t ♠♣♥ ssr ♥♥ t s t♥♦♦② ♥ t s♥r♦ s♣t♦♥s rtr

♣tr ♥tr♦t♦♥

t ♥♥ ♣r♠trs r ①trt r♦♠ t s♥r♦ t♦ t ♦t♦♥ ♦rt♠s ♠sr Ps ♦ t ♣♦♥ts ♦ t ♠sr♠♥t ♠♣♥ r ♦♠♣r t♦t♦s ♦t♥ t ♠t s♠t♦♥s ♥ t s ♦t♥ t t t♦♦r ♦♠♣r t♦ t ♠sr♠♥ts ♦r ♣♦♥ts ♥② t ♦③t♦♥ ♦rt♠sr t rst t ♣r♦r♠♥s ♦ t P ♦rt♠ r ♦♠♣r t♦ ♣♣r♦ ♦♥ t ♦ t st t♥ t ♠t♦ s ♦♥ ②♣♦tss tst♥ s♣♣ ♦♥ s♣ stt♦♥ s♥ r ♠sr♠♥ts

♥ t♦♥ t♦ ts ♦r ♠♥ ♣trs t ♥tr♦t♦♥ ♥ t ♦♥s♦♥ ♥♣♣♥① s ♣r♦ t t ♥ ♦ t ssrtt♦♥ s ♣♣♥① srs t t♠♥s♦♥ rr② ♦♠♣tt♦♥ rs s t♦ ♦♠♣t t t♦r③ ♥♥

♣tr

tr♠♥st Pr♦♣t♦♥♦♥ ❱t♦r③r♣s ♣♣r♦

♥tr♦t♦♥

♥ ♦♥t①t r t ♥♠r ♦ rss s ♥ t t rt ♦ t ♠r♥♥t♦r ♦♥t♥♦s② ♥rs t ♥♦ ♦ t ♣r♦♣t♦♥ ♥♥ ♥s t♦ ♠♦r ♣rs t♦ ♠♥ ♣r♦♣r② t r♦ rs♦rs ♥ t s♠ t♠ rsr ♦♥ t tr ♥t♦rs rs tt t ♦③t♦♥ ♥♦r♠t♦♥ ♦ ♦♠② ♦r ♦♣t♠③♥ t ♥rstrtr ♦♥sr♥ t♦s s♣ts t ♣♣rstt t s♠t♦♥ t♦♦s t♦ ♠r ♥ ♦rr t♦ t ♥t♦ ♦♥srt♦♥ ♦t♣r♦♣t♦♥ ♥ ♦③t♦♥ s♣ts

♦♥r♥♥ t ♣r♦♣t♦♥ ♥♥ ♣rt♥ t♦♦s t♦ ♦♣t♦♥s r ♥r② ♥s tr s ♦♥ sttst ♦r ♦♥ tr♠♥st ♣♣r♦s ♥ ♦③t♦♥ ♦♥t①t tr♠♥st t♦♦s r ♥r② ♣rrr t♦ ♣t② t♦ ♣r♦ st s♣♦srs rqr ♦r t♦♥ ♣r♣♦ss

t♦♥ ♣r♦s rst r s♠♠r② ♦♥ tr t③t♦♥ ♦ t♦♦ ♦r r♦♥♥ ♠♦♥ ♥ ♣rtr t ①st♥ t♦♦s r sss ♥ rr ♦ t s♣♥s ♦ t ♦③t♦♥ ♣r♦♠ ♥r ♦♥srt♦♥s ♦♥ t ♣r♦♣t♦♥ ♥♥r r♠♥ ♥ ♠t♠t ♠♦s ♥ tr ♠♥ rqr ♣r♠trs r t

♥ ♦rr t♦ rss t ♥♦♦r ♦③t♦♥ ♣r♦♠ ♥ ♠♦t② ts ♣tr ♣r♦♣♦sst sr♣t♦♥ ♦ ♥♦ t♦♦ ♥t♥s② s ♦♥ r♣ t strtrs sr♥ st♦♥ r② rt♦♥ ♦ t ♦♠♣tt♦♥ t♠ ♥ t ②♥♠ ♦♥t①ts rss t t ♣ ♦ t♦ ♣r♦♣♦s ♦t♦♥s rst ♦t♦♥ s ♥ ttr♠♥t♦♥ ♦ r②s t♠ss ② s♥ ♣rt ♦ts r② s♥trs♥ st♦♥ r♥ t r② tr♠♥t♦♥ ♣s s♦♠ s♥trs ♥ rs ♥♠♦♥ r♦♠ ♣♦st♦♥ t♦ ♥♦tr ♥trs r ♣rrs♦rs ♦ r②s r s♣t②♣rsst♥t tr sr♥ t ♣r♦r t♦ ♦t♥ r②s r♦♠ s♥tr ♥ st♦♥ t s♦♥ ♦t♦♥ s ♦♥ t t♦r③t♦♥ ♦ t t♦♥ ♦ t ♦♥ t

♣tr tr♠♥st Pr♦♣t♦♥ ♦♥ ❱t♦r③ r♣s

♣♣r♦

r②s t♦♥ srs t ♣r♦♣♦s t♦r③ ♣♣r♦ ♥s ♥ ♥tt♦♥ ♦r ❯❲ s♠t♦♥s ♦♠♣t ♠t♠t ♦r♠s♠ s ♦♥ t♠♥s♦♥ rr② t ♠♥s♦♥ rr② s ♥tr♦ ♦r tt ♣r♣♦s

♥ ♥t ♦ tt t♦r③ ♦r♠s♠ st s♣ ♠t ♠♣♠♥tt♦♥ s ♥tr♦ ♥ st♦♥ s ♣♣r♦ s ♠ s♠♣r ♥ ts str t♥t r② tr♥ ♦r t s ss t♦ ♥♦r♠t♦♥ ♦r ♦③t♦♥ ♦rt♠s t st♦♥ ♥s ♦♥ ♦♠♣rs♦♥ ♦ t ♥trt♦♥ ♦♥ts t♦♥t♦ ♣r①st♥ ♣♣r♦ s t♦♥ ♦ t ♦♠♣tt♦♥

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

s st♦♥ ♣r♦♣♦ss ♣♥♦r♠ ♦ t ♣r①st♥ s♠t♦♥ t♦♦s ♠t♦s ♥♥♦ ♦t t ♣r♦♣t♦♥ ♥♥ stt ♦ t rt ♦ t♦♦s s ♣r♦P♦♥t♥ t ♦ s♠t♦r t t♦ t ♦③t♦♥ ♥ r ♦ t rqr♠♥t ♦ s s♠t♦r q r♠♥r ♦ t ❯❲ t♥♦♦② s ♣r♦ ♥②♥rts ♦♥ t tr♠♥st ♥♥ ♠♦ r ♣r♦

♦♥ t Pr♦♣t♦♥ ♥♥ ♥ r ♦ t rr♥ts

❯ ♥ ♦s r♦♠ ③ t♦ ③ ♥ ts ♥ st s t♦② ♠♦st ♦t s rss st♥rs ♥ t r② ♦rs ♦ t ♦♣♠♥t ♦ t♦s st♥rstr s ♥ ♦rt ♦r ♣rt♥ t ♣r♦♣t♦♥ ♣r♦♣rts s ② t♦r ♦r ♦♦ s♥ ♦ rss ♥ ♦r ♥t♦r rst ♣♣r♦s ♥t♦ ♦♥t ♠sr♠♥t ♠♣♥s t r♦♠ t r② r② t♠ t s ♥ r②♠♣♦rt♥t t♦ t s♦ rt s♠t♦♥s r♦♠ t tr♦♠♥t t♦♦s

♦♠t♠s ♥♥rs ♥ ♠♦s ♣tr t ♠♥ trs ♦ ♥♥ ♦rs♥♥ s②st♠ ♥ s♦♠t♠s ♥♥rs ♥ t♦ s♣ ♥ ♥t t♦ ♣rtt s ♦ r q♥tts r ♣♦r P ♣t② ♦♥ ♥ st t ♥ ♣r♦♣rts t s t rs♦♥ ② ♦♥ ♥♦♥trs

sttst ♠♦s r ♠♦st ♦ t t♠ s♣② ♥ ♥ ♥st♥r s t♦ s♥ ♦r s♦♠ ♣r♣♦s

sts♣ ♠t♦s r ♠♦st ♦ t t♠ t t♦ t ♦♣rt♦r ♥s♦r ♦r ♥♦r♠t♦♥ ♦r t t♥♥ ♦ ♥ ♦♣rt ♥t♦r

ttst ♠♦s r t r♦♠ ♠sr♠♥t ♠♣♥s r ♣♦st ♣r♦ss♥♦s t♦ ①trt t ♠♥ rtrsts ♦ t ♣r♦♣t♦♥ ♥♥ ♦s ♠♦sr ♥r② s t♦ t t ♣r♦r♠♥s ♥ tr♠s ♦ t rt ♦r rr♦r ♣r♦t②♦♥ ♥ s♥r♦ ttst ♠♦s r r② ♦♥♥♥t s t s♠ st ♦♣r♠trs s ♣t ♦r r ♥♠r ♦ s♠r stt♦♥s ♦r ♥ ♥♦♦rstt♦♥ t ♥♥ s ② ♣♥♥t ♦♥ t ♥r♦♥♠♥t ♥ ♥ ♣rs♥t r②r♥t ♦r r♦♠ ♦♥ ♣ t♦ ♥♦tr ♥s r♦♦♠ ♥ ♥ ♦rr♦r♦r♦r t ♣r♦♠ ♦ ♥♦♦r ♦③t♦♥ rqrs t♦ ♦t♥ st s♣ ♦srs

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

♥s ts ♥♥ rt②♥ ♦r ♦③t♦♥ ♣r♣♦ss st s♣ ♠t♦ r ♥r② ♣rrr t

s♣ ♠t♦s r s♦ ♥♦♥ s tr♠♥st ♠t♦s ② ♦♣♣♦st♦♥ t♦ sttst♠t♦s tr♠♥st ♠t♦s r s ♦♥ t ♣r♦♣t♦♥ ♣r♦♣rts ♦ tr♦♠♥t s ♣♥♥ ♦♥ s♦t ♣r♣♦s tr ♣♣r♦s r ♥r② ♦♥sr

♥t r♥ ♠ ♦♠♥ ♣♣r♦s t♦ ♦ ♦♠♥ts ♦ r②s s ♣♣r♦s t♥q s ♥ ①t ♠t♦ t s s ♦♥ srt③t♦♥ ♦ t

①s qt♦♥s s♥ ♥trr♥ ♣♣r♦①♠t♦♥s t♦ t s♣ ♥ t♠ ♣rt rts rst♥ tr t♦r ♦♠♣♦♥♥ts r s♦ ♦r ♥♦♠ ♥ ♥ ♥st♥t ♦ t♠ ♣r♦ss ss t ♣r♦s tr♦♠♥t st♠t♦♥ t♠ ♥st♥t t− 1 ♦r ♦♠♣t♥ ♥ st♠t♦♥ t t♠ ♥st♥t ts♣t ♦ ts r r② t ♠t♦ rqrs ♣♦r ♦♠♣tt♦♥ t♦ ♣r♦rsts ♥ t ♣r♦♣t♦♥ ♦♥t①t s t ♥t s♦♥ ♠t♦ ♣r♦♣♦s ♥ ❬❪ s♥r② ♦♣ t♦ ♦tr ♣♣r♦s t♦ r t ♦♠♣tt♦♥ t♠ ❬❪ ♦r ♥♦t ①t rs♦t♦♥ ♦ ① qt♦♥ s ♥♦t ② s ♥ t ♣r♦♣t♦♥ ♦♥t①t♠♦st② s t t ♠tt♦♥s ♦♠s ♠♦r r♦♠ t ♥rt♥t② ♦ t r♦♥trt sr♣t♦♥ ♥r♦♥♠♥t t♥ r♦♠ t r② ♦ t ♠t♦ ts ♦rts ♠s s♥s ♥ ♦r ♥ t ♥ t ♦♠s t♦ rss rq♥② ♦ ❯s②♠♣t♦t ♠t♦s r r② ♠ st ♥ s♦ s ♦r ♦♥ t♠ st

♦ ♦♣ t ts ♦♠♣tt♦♥ ♦♠♣①t② t ♦ ♣♣r♦ ❬❪ ♥ ♥ss ♠t♦ rqrs t♥ ♦♥② ♦♥r② s rtr t♥ s tr♦♦tt s♣ s ♣♣r♦ s s♣② ♦♥♥♥t ♦r ♣r♦♠s ♥ t st ♥r♦♥♠♥t s ♥ t ♦rr ♦ ♠♥t ♦ t ♥t s ♣♣r♦ s ♥r②s t r② s ♣♣r♦s tr s t♦♥ ❬❪ ♦r ♦r ②r ♣♣r♦s ❬❪

r② s ♠t♦ r s ♦♥ ♦♠tr ♣ts ♥ ❯♥♦r♠ ♦r②♦ rt♦♥ ❯ ♦♥srs tt ♥ tr♦♠♥t ♥ ss♠t t♦ r② s r② ♥trts t r♥t ♠♥t ♦ t ♥r♦♥♠♥t ② ♥rt rrt ♦r rt ❯ ♦s t♦ r♠♥t t s♦♥t♥ts♦ t t ♥♦r♠ rt♦♥ ♦♥t ❬❪ ♦s ♦♥trt♦♥s r s♣②♠♣♦rt♥t ♥ r♥ ♦t♦♦r stt♦♥s

❯♥ t ♣r♦s ♠t♦ t r② s ♠t♦s ♥♥♦t s ♦r ♦ rq♥s t②♣② ♦ 100 MHz ♥ ♥ ♥trt♥ ♦ts ♥ t ♥r♦♥♠♥tr s♠r ♥ rr ♦ t ♦♥sr ♥t ♦♣② t♦s t♦ ♦♥t♦♥s♦♠s ss rstrt ♦♥sr♥ t ♥rs ♦ s rq♥② ♥ ♦♠♠♥t♦♥st♥rs

② tr♠♥t♦♥ ♠t♦s r ♥r② ♥ sr♦♣s ②r♥ ♥ r② ♥♥

r② ♥♥ ♦♥ssts ♥ ♥♥ r②s r♦♠ t tr♥s♠ttr ♥ ♥ t♦s♦ r ♣r♥ ♥t② ♦ t rr

♦♥ssts ♥ tr♥ r②s r♦♠ t rr t♦ t tr♥s♠ttr ①♣♦r♥t ♦♠♥t♦r② ♥ ② t♠ ♦♥s♠♠♥


♣♣r♦

♥t ♦ ②r♥ ♦♦s

r② tr♥ t♦♦s ♥ ② s ♦r r♦ ♦r ♥ r♥ ♥r♦♥♠♥tr♥ t ①♣♦s r♦t ♦ t r t♦♠♠♥t♦♥ ♥str② ❬❪❬❪ ❬❪ ❬❪❬❪ ❬❪ ♥ ♠♣♦rt♥t ♠st♦♥ ♦r s♥ t♦♦ ♦r ♥♦♦r ♣r♦♣t♦♥ s ♣r♦②♥ r t rst ♦♠♣rs♦♥ t♥ r② tr♥ s♠t♦♥ ♥ ♥♦♦r ♠sr♠♥ts r r♣♦rt ❬❪ t♦ r② rt ♦♠♣rs♦♥ t t ♠sr♠♥tst ♠t♦ ♣r♦♦ ♥trst ♥ s♣② ♥ tt ♣r♦ ♦ t ♥rs ♦r♦ ♦♠♠♥t♦♥ ♥ s♣② ♥ ♦♥t①t ♦ r t♦♠♠♥t♦♥ ♠r♥ t t ♥ ♦ t s t ♠t♦ ♦♥t♥ t♦ ♠♣r♦ ♥♦t② ♥ r♥t s♠♥ ♦rs s♣② ♦♥r♥♥ t ♠♣♠♥tt♦♥ ♦ r♥t ♥ ♥ t❯♥♦r♠ ♦r② ♦ rt♦♥ ❬❪ s ♦r♥r st♦♥ ♦ t♦s ♠r♥ tr♠♥st ♠♦s t tt t♠ t t♦r② ♦ rt♦♥ s s♦ ② s ♥ sttsts ♦r r r♦ss t♦♥ rt♦♥

❯s♥ t♦s ♠♣r♦♠♥ts t s♣♣♦rts t ♦♠♣rs♦♥ t ♥ ①t ♠t♦s ♦ ❬❪ t♦ r r② ♠ str t♥ ①t ♠t♦s tr s♣s ♥ ♥rs s♥♥t② ♥ ② ♦♥sr♥ t ♣rt♦♥ ♦ st②rt♦♥s ❬❪ t ♣r♣r♦ss♥ ♦s t♦ ♥ 103 t♦r ♦♥ ♥♦r♠ ♥ ts♥s t ♣♦sst② t♦ rss ♠♦r ♦♠♣① s♥r♦ s ♦♠♣t t② ♦r ♦r♥♦♦r ♦r

s♠ ②r t ♦♠♣t ♦r ♦ ♥ r♠♥② s ♦♠♣r ♥stsr ♠sr tsts ♥ ts ♠♦♥strt tt r②s s♠t♦♥ ♦rs t② ♣♣r♦ ♦r ♣rt♦♥ ❬❪ ♥ t s t ♠♥ ♦ r t♦♠♠♥t♦♥ st q② ♥rss ♥ rqrs t rrr ♦♣rt♦r t♦ q② ♦♣tr ♥t♦rs ♦r tt ♣r♣♦s ♦♠♠r t♦♦s ♣♣rs r♥ ts ♣r♦ ss ❱♦♥♦ ❬❪ ❲♥Pr♦♣ ❬❪ ❲rss ♥st ❬❪ ♦r ♥t ❬❪

rr♥t r♥s ♥

♦ s♣♣♦rt t ♦♥♦♥ ♦t♦♥ ♦ r ♥t♦rs ♥ tr♦♥♦s ♦♥rt♦♥s r♦st♥ ♥t♦rs t s ♥ssr② t♦ ♣t t t♦♦s ♦r♣rt♥ t ♦r ♥ ss♦t ♣♣t♦♥ srs s s ♥ ♦♥ ♦♥ t t♦ s ♦ ts st♦r② r♥ ts s ♦t t ♥t♦rs ♥♣♣t♦♥s ♦ ♦② t ♥t♦rs ♥rs ♥ tr♠s ♦ rq♥②♥ ♥t ♥ ♦rr t♦ ♣r♦ r t rts ♦r t ♥♠r ♦ ♥t♦rss s♦ ♥rs t♦ ♠♦r s♣ ♥ rrs ♦ t rqr♠♥ts ♦ t srs ♥♥ ♣r t srs ♦♣ ♥ ♥s rt t♦ t ♥t♦rs ♣♦ssts

r♦♠ t♦s s♣ ♥s ♦③t♦♥ s ♣rtr② ♥trst♥ ♣♣t♦♥ s s tr t♦♥ ♦③t♦♥ ♥ ♣♦♠♥ ♥t♦rs ♦ st rqr♠♥t ♥ ♦r t sr ♥ ♥ ♦r t ♥rstrtr ts

Pr♦①② ♦♥ t ②rs ♦ ♦♣♠♥t t♦♦s ♠♦st② r♠♥t♦♦s ♦r ♣rt♥ t ♣r♦♣t♦♥ ♥r ①♣♦tt♦♥ ♦ t♦♦s ♦r tst♥♥♦♦r ♦③t♦♥ ♦rt♠s s r♥t② ♥ ♥♦t ♥ ❬❪ rtss t♦♦srqr t♦ ♣t ♥ ♦rr t♦ rss t ♣♦♠♥ ♣r♦♠t ♦ ♦③t♦♥♦r ♦♦♣rt♦♥ ♥ ♥s t♦ ♣rt ♠♦r ♥ ♠♦r ♥t♦rs ♦t ♦ t

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

t♦♦s ❬❪ t t ♦♥r♥ ♦ t s♣ ♥ ♦ t ♦③t♦♥ ♥ t♦ t tr♥♦ ♥rs♥ rq♥② ♥ ♥t♦rs ♣♣rs t ❯❲ t♥♦♦②

r♥② rsr t♦ t ♠tr② s t ❯❲ ♥t s ♥ r ♥♥s t♦ s ② t ♦♥ t rr② ❬❪ ♦r ♥ r♦♠ ③ t♦③ s rs ♦ s♣tr♠ ♠♣t ♦ t s♥t ♦♠♠♥t②t t ♣rs♣t ♦ s s♣tr♠ ♦♣♣♦rt♥ts ♦r s♦rt r♥ ♦♠♠♥t♦♥ ♦t ♦r ♦ ♥ t rt t tt t♥♦♦② s q② tr♥r♦♠ ♦♠♠♥t♦♥ ♣♣t♦♥ t♦ ♦③t♦♥ ♣♣t♦♥s t♥s t♦ t ♥t♠tr r② ♥ r♥♥ ❬❪

♥ t♦♥ t♦ tt ♥trst♥ r② ❯❲ t♥♦♦② ♣r♦s ♦tr ♥♦r♠t♦♥ ♦♥ t ♣r♦♣t♦♥ ♥♥ s rt s♥ ♣ts ❬ ❪ t♦♠♦st t♦♦s s♥ s r ♥t r s ♦♥ ①t ♠t♦s ❬❪ ♦♥② t♦♦s t ♥ ♦♥srt♦♥ tt t♥♦♦② ♦r tt ♣r♣♦s ♠♦r ts ♦♥ ts♣ts ♦ t ❯❲ s♥ r ♣r♦♣♦s ♥ sst♦♥

♥♦tr ♥r②♥ s♣t ♦ t ♦③t♦♥ ♣r♦♠ s t ♠♦t② ♦ t ♥♦s♥ ♥t♦ ♦♥t t ♠♦t② ♦ t ♥♦ ♥ t♦♦ s ♥r② ♦♠♣① ts♥ t ♠♦♠♥t ♦ t ♥♦s rqrs t♦ r♦♠♣t t t♠ st♣ ♥ t♦ t♦ st♠t ♦rrt Ps s♣t t r♣r♦t② ♦ t ♣r♦♣t♦♥ ♥♥♦s t♦ ♦♠♣t ♦♥② ♦♥ t♠ t t♥ t♦ t♦ s t ♦♠♣tt♦♥t♠ ♦♠s ♦tt♥ ♥ ♣r♦♠s t r ♥♠r ♦ ♥♦s r ♦♥sr ♦♠♣①t② ♦ t ♣r♦♠ ♦s ♥♦t t t t ss r②♥♥ r②tr♥♣♣r♦ s ♦t ♥♦t q ♥♦ ♥ ♥♦t ① ♥♦ t♦ rss tt ♥♦ ♣r♦♠s ♦ ♦♣ t tt ss s♦t♦♥ s t♦ ♥s ♥ ♥r♠♥t t♦♦st♦ ♦ rstrt♥ t r② ♦♠♣tt♦♥ r♦♠ t ♥♥♥ ♦r ♣♦st♦♥ ❬❪

♥② t ♣r♦♠ s ♥ ts tss s t ♦♦♥ ♦ t♦♦s t♦ s♥ t♦ t ♥t♦ ♦♥srt♦♥ ♦③t♦♥ ♣r♦♠ s ♦♥ ❯❲t♥♦♦② ♥ ♦tr ♦rs ♦ ♥ ♣r♦ st ♥ rt ♥♦r P♦r r ♥t

♣ts ♦ t ❯tr ❲ ♥ ❯❲ ♥♦♦②

♠♦♥ ♦tr t ♣r♦s st♦♥ s ♣rs♥t q ♦r ♦ t t♦♦s t ♦r ♦③t♦♥ ♣r♣♦ss t ♣♣r tt t ❯❲ t♥♦♦②♣rs♥ts sr ssts ♦r ♦③t♦♥ ♣r♣♦ss tt r ♦♥ t♦ ♣rs♥t ♥ tssst♦♥

♥rts ♦♥ ❯tr ❲ ♥ ❯❲

♥ ❯❲ s♥ s ♥ s s♥ t rt♦♥ ♥t B

B =fh − fl

(fh + fl)/2> 0.2

t fh ♥ fl r rs♣t② t rq♥② ♥ ♦ rq♥② ♦ t ♦♥sr♥ ♥ t −10 dB s s♦ ♦rrs♣♦♥s t♦ ♥② s♥ t ♥t rtr


♣♣r♦

♦r q t♦ MHz ♥ tt t ♥♥ ♣t② C ♦ s②st♠ ♥ ❲♦♥t♦♥ s rt t♦ ts ♥t W ♥ SNR s

C =W ♦2(1 + SNR)

SNR =PrN0W

❲r Pr s t r ♣♦r ♥ N0 t ♥♦s ♣♦r s♣tr ♥st② t r②♣♣rs tt ❯❲ t♥♦♦② ♥ ♣♦t♥t② ♦rs r② t rt t ♥ s♦♦r t♠ rs♦t♦♥ ♥ s♥♥t r♥ t ♦ t rt s st ♣r♦♣rt②♠s ❯❲ ② t♥♦♦② ♦r r♥♥ ♣♣t♦♥s tr② t tr♥s♠tt♣♦r s t♦ rt ♥ ♦rr ♥♦t t♦ ♥trr ①st♥ t♠t s②st♠s ♥ ts t♥ ♥rst♥ tt t st rs♥ ♥t♦♥ ♦ r♥ ♥ ♥♥s t ❯❲ ♦r t②♣s ♦ ♣♣t♦♥ rr ♠♥ ♥♠♦♥t♦r♥ ♠ ♠♥ ♠♥ tr♦ ♦sts ♥ ♦♠♠♥t♦♥ s②st♠♥ ❯ t ♦r♥ ♠ss♦♥ ♠s ♥ ♥ r ♥ r♦♣ t♦ ♣r♥t♥trr♥s t ♣r①st♥ st♥rs t r♦♣ ♥ r ♦ ts rt♦♥s ♠♣♦s ♠♦r rstrt ♦♥t♦♥s ♦♥ t ❯❲ ♠ss♦♥ ♠s s t ♥ s♥♦♥ r

❲♥ ♦♦♥ t ♠♣♦s rt♦♥ t ❯❲ t♦rt② s t ♥tt♦ ♣rs♥t r② ♦ ♣♦r ♥st② s♣tr♠ ♦sts ♦①st♥ t ♦trs②st♠ ♥sr♥ r② ♦ ♥trr♥s ♦r♦r t s ♦ r ♥ s t♥t t♦ ♠t t st ♥ ts ♥ ♥r♦♥♠♥t ♥rt♥ ♥♠r ♦♠t♣ts ♥♦♦r ♥r♦♥♠♥t ♥ ♥t♦♥ ts ♥ rs♦ s ♦♥ s r♣t♦♥ s s♣② s♥ t♦ t ♥t ♦ t♦s ♠t♣tr rr ♥♦tr ♥trst♥ ♣r♦♣rt② ♦ ❯❲ s♥s s tr ♦♦ ♠tr♣♥trt♦♥ ♣ts s♣② ♥ t ♦st ♣rt ♦ t s♣tr♠

Frequency (GHz)

dBm

/MH

z)

♠s

Frequency (GHz)

dBm

/MH

z)

♠s

❯❲ t♦♠♠♥t♦♥ ♠ss♦♥ ♠s ♦r ❯ ♥ r♦♣

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

❯tr ❲ ♥ ❯❲ ♠♣s ♦ ♦r t♥ r♥t♣♣t♦♥s

♥ ♥ t ss♦t♦♥ ♦ t st♥r③t♦♥ r♦♣ t♦♠♥ t②♣s ♦ ❯❲ t♥♦♦s r r♥tt

t♥ ❯❲ r s♥s r ♠tt ♦♥ ♠t♣ rq♥② rrrs t♥t > MHz

♠♣s s♣♦♥s ❯tr ❲ ♥ ❯❲ ♦rrs♣♦♥s t♦ s♦rt♠♣s ♠ss♦♥ ♥ t ns ♦rr ♦ ♠♥t t ♦r t♦t ♥tr rrrrq♥②

t♥ ❯❲ t♥♦♦② s ♥r② ♥s ♦r rt ♥ s♦rtr♥ ♣♣t♦♥s t♥ ♥t ♦ t ♣t② ♣r♦♣rts ♦ t ❯❲ s♥st♦ s♦♠ ♠r♥ ♦♠♠r ♣♣t♦♥ t t ❲rss❯ t s♦ ❯❲ ♦r ♦♠♠♥t♦♥ s ♥r r② ♦♥♥ ss♦t♦♥ ♦ rst t st♥r③t♦♥ r♦♣ ♥ s♦♥ t ❲ ♥ s ♠tt ♦♣♠♥t ♦ t ❯❲ t♥♦♦② ♦r ♦♠♠♥t♦♥ ♥♦②s s ♠♦r♦♥r♥ ② t ♠r♥ GHz t♥♦♦s

❯❲ t♥♦♦② s ♥r② s ♦r ♦③t♦♥ ♣r♣♦s ♥ st ❯❲ ♥srs t♦ ♥rt s♠ ♠♣s♦♥s ♥ t♠ ♦♠♥ t ♦s t♦ ♣♣rt②s t ♣rs♦♥ t t r♣t♦♥ rst s ② t ♠tr② ♦r rr ♣♣t♦♥ t GHz t ♦③t♦♥ ♣t② r ♦r t♦ rr ♣ r♦♠ t st♥r③t♦♥s ♥ ①♠♣ ♦ ts st♥r s t ❬❪ s ♥♠♥♠♥t ♦ t ♦ t rt ❩ st♥r s t ♥t t♦ ♦r r② ♦ ♦♥s♠♣t♦♥ ♥ ♦s t♦ ♦♥♥t r ♥♠r ♦ rssq♣♠♥ts ♦r ♠t ♦st ❬❪ s ♦♥ ts st♥r ♦♠♠r ♦♠♣♥s ♣r♦♣♦s ♠♣r♦♠♥t ts s s♣② t s ♦ ♣♦♦♥ r♥t② st ♦r r♦r ♦ r♥ ♠sr♠♥t ❬❪ ❬❪ ♣r♦s s♦ ♥ ♠♣ ♥♥ ♥ rt ♣♦st♦♥ st♠t♦♥ ♥♦tr ♠r♥ st♥r st s♣② t ♥ ♣♣t♦♥ t r② ♦ t♥② s st♥r ss♣② t ♦r ❲rss ♦② r t♦rs ♣♣t♦♥s ❬❪

♥ s♣t ♦ tr r♥s t♦s st♥rs ♦♦s② r② ♦♥ tr♦♠♥t ss s♣♣♦rt ♦ t ♥♦r♠t♦♥ ①♥ ♥ t ♦♦♥ s♦♠ ♠♣♦rt♥t s♣ts ♦♥t ♣r♦♣t♦♥ ♥♥ r r♠♥

♥rts ♦♥ t Pr♦♣t♦♥ ♥♥

s st♦♥ r s♦♠ ss ♦t ♦ tr♦♠♥t s ♣r♦♣t ♥♦ t② ♥trt t t ♥r♦♥♠♥t

♦♥ t tr♥s♠ss♦♥ ♥♥

♣r♦♣t♦♥ ♥♥ s t s♣ r t tr♦♠♥t s r ♣r♦♣t tr♥s♠ss♦♥ ♥♥ ♥s r②t♥ t♥ t tr♥s♠tt s♥ ♥r s♥ ♥♥ ♥t♥♥s ♥ ♦♠♣♦♥♥ts ♦ ♦t tr♥s♠ttr ♥ rrs


♣♣r♦

tr♥s♠ss♦♥ ♥♥ ♥ s♥ s t♠ ♣♥♥t ♥r tr tr♥s♦r♠♥♥ ♣♣ ♥♣t tr s♥ t♦ ♥♦tr ♦♥ ② tt♥t ♥ ♦r st♦rt

ss② ♦r♥ t♦ ♦ ❬❪ t ①sts q♥t r♣rs♥tt♦♥s ♦ t tr♥s♠ss♦♥ ♥♥ r♣rs♥t ♥ r

♠r♥t tr♥sr ♥t♦♥ h(τ, t) rt② r♣rs♥t t ♥♥ ♠♣s rs♣♦♥s ♦ t ♥♥ ♦t♦♥ s♥ y(t) s ♦t♥ r♦♠ t ♦♥♦♥ s♥ x(t) s♥

y(t) =

∫ +∞

−∞x(t− τ)h(τ − t)dτ

r h(τ, t) s t ♥♥ ♠♣s rs♣♦♥s ♦ t ♥♥ t t♠ t ♦r ♣st♥t t τ

rq♥② r♥t tr♥sr ♥t♦♥ T (f, ν)ss♠♥ tt X(f) ♥ Y (f) r t ♦rr tr♥s♦r♠ ♦ x(t) ♥ y(t) rs♣t② t s ♣♦ss t♦ rt

Y (t) =

∫ +∞

−∞X(f − ν)T (f, ν)dν

r T (f, ν) ♥ ♦t♥ r♦♠ t ♦rr tr♥s♦r♠ ♦♥ ♦t t ♥ τ s s

T (f, ν) =

∫ +∞

−∞

∫ +∞

−∞h(t, τ)e−j2πfτe−j2πνtdτdt

♠ rq♥② ♥t♦♥ H(f, t)s ♥t♦♥ ♦s t♦ ①♣rss t rq♥② rs♣♦♥s ♦ t ♥♥ s ♥t♦♥♦ t♠ ♥ t ♦t♦♥ t♠ s♥ y(t) ♥ ①♣rss r♦♠ t ♥♦♠♥rq♥② s♥ X(f)

y(t) =

∫ +∞

−∞X(f)H(f, t)ej2πftdf

❲r H(f, t) ♥ ①♣rss t

H(f, t) =

∫ +∞

−∞h(τ, t)e−j2πfτ =

∫ +∞

−∞T (f, ν)ej2πνtdν

②♦♣♣r ♥t♦♥ S(τ, ν)s ♥t♦♥ ♦ t♦ ①♣rss rt♦♥ ♦ t ♥♥ s ♥t♦♥ ♦ t ♦♣♣rss♠♥ tt ①♣rss♦♥

y(t) =

∫ +∞

−∞x(t− τ)

(∫ +∞

−∞S(τ, ν)ej2πνtdν

)

dτ

①♣rss tt ♣rt ♦ t r s♥ s ② rs♦♥ ♦ t♠tt s♥ t s ♣♦ss t♦ rt

S(τ, ν) =

∫ +∞

−∞h(τ, t)e−j2πνtdt =

∫ +∞

−∞T (f, ν)ej2πfτdf

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

♦ ♥t♦♥s sr♥ t ♣r♦♣t♦♥ ♥♥ ♥ ♦♥♥t♦♥ tr t♥

♥ s ♥② ♠♦t② s ♥t t ♥♥ s s stt t s t♥ ♣♦ss t♦r♠♦ ♣♥♥s t t♠ ♥ ♦♣♣r rq♥② t qt♦♥s ♥ rt♦ ♦♥② t ♥♥ ♠♣s rs♣♦♥s h(τ) ♥ t tr♥sr ♥t♦♥ ♦ t ♥♥H(f)

Pr♦♣t♦♥ ♦sss

♥ r s♣ t r ♣♦r Pr t st♥ d ♥ ①♣rss s♥ t rs♦r♠ ❬❪

Pr = PtGrGt

(

λ

4πd

)2

r Pt s t tr♥s♠tt ♣♦r Gr ♥ Gt r t ♥t♥♥ ♥ ♦ t rr♥ t tr♥s♠ttr ♥ λ s t ♥t t ts ♣♦♥t t s ♣♦ss t♦ ♥t ♣t ♦ss ♥ r s♣ s t rt♦ t♥ t tr♥s♠t ♥ t r ♣♦r♥t♥ t t♦ ♥t♥♥s s ♣t ♦ss L s s② ①♣rss ♥ dB t

LdB = −20♦10(

λ

4πd

)

Prt② ts qt♦♥ s ♥r② ①♣rss rt② t♦ rr♥ st♥ d0s

L = L0 + 10np♦10

(

d

d0

)

t

L0 = 20♦10

(

4πd0λ

)

♥ np s t ♣t ♦ss ①♣♦♥♥t np = 2 ♦r r s♣ stt♦♥♦r ♥♦♥ r s♣ ♣r♦♣t♦♥ ♥♥ t ♥♥ ♦ t ♥r♦♥♠♥t s t♥

♥t♦ ♦♥srt♦♥ ② ♥tr♦♥ ♣t ♦ss ①♣♦♥♥t ♣r♠tr np ♥ t ♣♦ss


♣♣r♦

♦sts ♣r♦♥ rt♦♥ sttr♥ ♥ rt♦♥ ❬ ❪ r ♠♦ t ss♥ r♥♦♠ r Xsh t st♥r t♦♥ σsh r♥♦♠ rXsh ♠♦s t s♦♥

L = L0 + 10np♦10

(

d

d0

)

+Xsh

s ♦r♠ ①♣rss rt♦♥ t♥ st♥ ♥ ♣r♦♣t♦♥ ♦sss t ♥ ①♣♦t ♦r rt♥ r ♣♦r t♦ st♥ ♦r r♥♥

♦♥ t t♣ts

♥ s♦♠ r♠st♥s t tr♥s♠ttr ♥ t rr ♥ ♥ rt st②s stt♦♥ s ♥ ♦ t stt♦♥ ♦r t ①sts r ♥♠r♦ stt♦♥s r t rr ♥ t tr♥s♠ttr r ♥ ♦♥ ♥ ♦ t s stt♦♥ ♥r② ♣♣♥s ♥ ♦sts r ♥srt t♥ ♦t ♥s ♦t tr♥s♠ss♦♥ ♥♥ r s♥ s t♥ s♠ ♦ ♦s tt♥t ♥② ♥ t♠ ♥ st ♥ ♣s ♦r s♥ s♥ r♣rs♥tt♦♥

s r♣rs♥t ♥ r tr ♠♥ ♣♥♦♠♥ tts ♦ s r ♣r♦♣t ♦ ♦sts

t♦♥ s♣r ♦r s ♥ rt♦♥s ♣♥♦♠♥ ♣♣♥s ♥ t rrs t t ♥tr ♦ ♠trs s♦♥ s t rrrt② ♦ t ♠trs r s♠r ♥ rr ♦ t ♥t ♥s r t ♠tr s ♦♠♣t② rt♥ t ♥tr ♥♦♠♥ ♥r② ♦ t s rt ♦♥ t ♥tr s ♣♥♦♠♥ s s♣r rt♦♥♥ s r t ♠tr s ♥♦t ♦♠♣t② rt♥ ♣rt ♦ t ♥r② ♦ t♥♦♠♥ s rt rs ♥♦tr ♣rt s rrt ♥t♦ t ♠tr ♣r♦♣♦rt♦♥ ♦ rt ♥ rrt ♥r② s rt② rt t♦ t ♣r♦♣rts ♦t ♠trs t rrrt② ♦ t ♠trs r ♥♦t ♥ t rt♦♥s s ♠♥s tt t ♥r② s s♣rs ♥ ♠t♣ rt♦♥s

r♥s♠ss♦♥s ♦♥ s t ♦sts ♥t t♥ss t rrt ♣rt ♦ ♥ ♥♦♠♥s ♥ ② ♦s tr♦ t ♦st ♥ tt s ♥ t rrt ♣rt♦s ♦t r♦♠ t ♦st s♦♥ rrt♦♥ ♣♣rs ♠t♣ rrt♦♥♣♥♦♠♥ ♥♥ ② r♦ss♥ ♠tr s tr♥s♠ss♦♥

rt♦♥rt♦♥ ♣♣♥s ♥ ♥♦♥trs t ♦ ♥ ♦st r ♥rr ♦ t ♦♥sr ♥t ♦r♥ t♦ t ②♥s ♣r♥♣ trt♦♥ ♣♦♥t rts ♥r② ♥ t rt♦♥s ♦ s♣

t♦s ♥trt♦♥s rt ♠t♣t ♥ t ♥♥ ♥ t s s t♦ rtt stt ♥♥ ♠♣s rs♣♦♥s h(τ) s s♠ ♦ t♦s r♥t ♣ts s

h(τ) =

K−1∑

k=0

αkδ(τ − τk)

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

Specular Reflection Reflection and Refraction

TransmissionTransmission Diffraction

Diffuse Reflection

♥ ♣r♦♣t♦♥ ♣♥♦♠♥

r αk ♥ τk r t ♠♣t ♥ t ② ♦ ♥ ♣t k ♥ δ rrs t♦ tr t strt♦♥

t ♦♥ t ♥♥

②♣② t ♥♥ ♦ t r♥t ♣♥♦♠♥ rt t♦ t ♠t♣ts s♥ ♥♥ ♦♥ t r s♥ ♦ ♣rtrt♦♥s ♥ ♦sr t♦ t♦s♣♥♦♠♥s

s♦ s♦♥ t t ♦rrs♣♦♥s t♦ s♦ t♠ rt♦♥s ♦ t r s♥ t♦ t t♣♦st♦♥ ♦ t ♦sts t♥ t tr♥s♠ttr ♥ t rr s tt②s ♥r② ♠♦ ② ss♥ r♥♦♠ r

st s♦♥ t s♦ ♥♦♥ s ♥ t ♦rrs♣♦♥s t♦ st t♠ rt♦♥ ♦ t r s♥ t♦ t ♦♥strt♥ strt s ♣r♦ss ♦s tts r ♥r② ♠♦ rs♣t②② ② ♦r ♣♥♥ tr ♣r♦♠♥♥t ♣t ♥ ♥t ♦r ♥♦t

♣ts ♦ t ❯❲ Pr♦♣t♦♥ ♥♥

t♦ ts r ♥t t ❯❲ ♣r♦♣t♦♥ ♥♥ ♣rs♥ts s♦♠ ♣rtrts t♦ t♥ ♥t♦ ♦♥t t♦ rst ♠♦ ♠♥ ❯❲♣r♠trs ♥ t s sttst ♠♦s r r♠♥ ♥ t ♦♦♥

Pr♠trs ♦r rtr③t♦♥

♦r♥ t♦ s♠ ❬❪ ♥ ♦rstr ❬❪ t ♣r♠trs t♦ t♥ ♥t♦ ♦♥t♦r ♥ ❯❲ ♥♥ ♠♦ r


♣♣r♦

♠t♣t ② s♣r r ♥r② s♠♠r③ ② τm t ♠♥ ② t♦ r t rr τRMS t st♥r t♦♥ r♦♥ t ♠♥ ② τmax t ② ♦ t st ♣t

♥♠r ♦ ♠t♣t ♦♠♣♦♥♥ts P ② t ♥♥s ♥♠r s t r♦♠ t ♠①♠ tt♥t♦♥ trs♦ tr♠♥r♦♠ t str♦♥st ♣t

♠t♣t ♥r② ♠♦♥t ♥♠r ♦ ♠t♣ts rqr t♦ t rt♥ ♦ ♥r②

♠t♣t ♠♣t ♥ strt♦♥qr ♦r ①trt♥ t ♣t♦ss ①♣♦♥♥t ♥ t st♥r t♦♥ ♣r♠trs

ttst ♦s

t s ♥ s♥ ♣r♦s② tt ♥ s ♦ ♥♦♦r ♣r♦♣t♦♥ r ♥♠r♦ ♠t♣ts ♣♣rs r♦♠ t t ♠♥s tt t ♦r♥ ♠tt s♥ stt♥t ♥ s♣r ♥ t♠ ♦r t ♣♣rs tt t ♠♦ ♥ ♥ ♦s♥♦t t ♥t♦ ♦♥srt♦♥ t t tt t r♥t ♠t♣t ♦♥sttt♥ t rr ② strs r♦♠ tt ♦srt♦♥ ♥ ❱♥③ ❬❪ ♣r♦♣♦s ♠♦ s ♥ ♣t ♦r ❯❲ ♥ t ♥ st♥r❯s♥ ts ♠♦ t ♥♥ ♠♣s rs♣♦♥s h(τ) ♥ rtt♥

h(τ) = AL∑

l=0

K∑

k=0

αk,lδ(τ − Tl − τk,l)

r A s ♦♥♦r♠ r♥♦♠ ♠♣t L ♥ K s♥t t ♥♠r ♦ ♦♠♣♦♥♥ts ♣r strs ♥ t ♥♠r ♦♦sr strs rs♣t②

Tl s t ② rt t♦ t lt str τk,l s t ② ♦ t kt ♦♠♣♦♥♥t ♦ t lt str αk,l s t ♦♥t ♦ t kt ♦♠♣♦♥♥t ♦ t lt str

str ② ♥ ♣t ② strt♦♥s ♦ ♥ str l rs rs♣t②

p(Tl|Tl−1) = Λe−Λ(Tl−Tl−1) ♦r l ≥ 1

p(τk,l|τk−1,l) = λe−λ(τk,l−τk−1,l)

t Λ ♥ λ r t ♣♣rt♦♥ rts ♦ t strs ♥ t♦ ♦♠♣♦♥♥t ♥t♦ ♥str rs♣t②

s t t ♦♥t ♥ rtt♥

αk,l = pk,lǫlβk,l

r

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

pk,l s t ♣♦rt② ♦ t s♥ ǫl ♥ βk,l ♣r♠tr③♥ t ♥ ♦ ♥ ♦♠♣♦♥♥t k ♥ str l r

log10(ǫlβk,l) ∝ N (vk,l, σ1 + σ2)

vk,l =10ln(Ω0)− 10Tl/Γ− 10τk,l/γ

ln(10)− (σ21 + σ22) ln(10)

20

E[|ǫlβk,l|] = Ω0e−Tl/Γe−τk,l/γ

σ1 ♥ σ2 r t st♥r t♦♥ ♦ t ♦♥♦r♠ ♣♦r tt♥t♦♥ rs♣t②♣r str ♥ ② ♣t ♥t♦ str Ω0 s t ♥r② ♦ t rst ♣t ♦ ♥str Γ ♥ γ r t tt♥t♦♥ ♦ t str ♦r ♦ t ♣t ♦ ♥ str t♦s r ♥ ♦sr ♦♥ r

②♣ ♥♥ ♠♣s rs♣♦♥s ♦r t ♠t♣ strs

♠♦ s r♦t s♦♠ ♠♦t♦♥s t♦ ts ♠♦ ♥ ♦rr ♠♦rrt ♦♥ t ♣♣rt♦♥ rq♥② ♦ t strs ♥ t ♣ts ♥ ♣rtr

t strt♦♥ ♦ t ♣ts t♠♦rr ♣r str p(τk,l|τk−1,l) s ♥♦ ♠♦ ② P♦ss♦♥s ♣r♠tr③ ② λ1 ♥ λ2 ♥ ♣r♠tr µ

p(τk,l|τk−1,l) =µλ1e−λ1(τk,l−τk−1,l)

+ (µ− 1)λ2e−λ2(τk,l−τk−1,l) ♦r k ≥ 1

♠♣t ♦ ♦♠♣♦♥♥t ♣r str ♣♥s ♦♥ t stt♦♥

E[|αk,l|2] = Ωl1

γl((1− ν)λ1 + νλ2 + 1)e−τk,l/γl

♦r stt♦♥ t

E[|αk,l|2] = 1 + χe−τk,l/γre−τk,l/γdγr + γdγd

Ωlγr + γd(1− χ)

r Ωl s t ♣♦r ♥t♦ t lt strγl ♥ ♥trstr rs t♦r χ ♣r♠tr sr♥ t tt♥t♦♥ ♦ rst ♦♠♣♦♥♥t ♥ γr ♥ γd ♥t♦r ♦♥ t♣♦r rs s♣ ♥t♦ str


♣♣r♦

tr♠♥st ♦ ♦ t Pr♦♣t♦♥ ♥♥

s st♦♥ strts t s♦♠ ss t♦r② ♦♥ t tr♦♠♥t ♣r♦♣t♦♥ ♥ ss♠♥ ♥ ❯❲ ♣r♦♣t♦♥ ♥♥ ss♠♣t♦♥s ♦ t ♠♦ ♥♥trt♦♥ ♦♥ts r r♠♥

①s qt♦♥s

♥② ♠♦♥♦r♦♠t tr♦♠♥t t ♣st♦♥ ω s ♠ ♦ ♥ tr E(s, ω) ♥ ♠♥t H(s, ω) ♦r ♥ ♦srt♦♥ ♣♦♥t ♥ ② tt♦r s ss♠♥ tt ♦t ♣r♦♣t ♥t♦ ♥ ♥♦♠♦♥♦s ♥r♦♥♠♥t t ♥ ♣r♠tt② ǫ(s) ♥ ♥ ♣r♠t② µ(s) t ♦r ①s qt♦♥s r

∇×E(s, ω) + jωµ(s)H(s, ω) = ~0

∇×H(s, ω)− jωǫ(s)E(s, ω) = ~0

∇ [ǫ(s)E(s, ω)] = ~0

∇ [µ(s)H(s, ω)] = ~0

♥ t ♦♦♥ t s♠♣ s ♦ ♦♠♦♥♦s tr ♥r♦♥♠♥t s♦♥sr s ♦♥sq♥ ♦t t ♣r♠tt② ♥ t ♣r♠t② r ♣♦st♦♥♥♣♥♥t s t♦ t s♠♣t♦♥ ǫ(s) = ǫ ♥ µ(s) = µ

❯s♥ tt s♠♣t♦♥ ①s qt♦♥s ♥ rrtt♥ s t t♦r③♠♦t③ qt♦♥

∇U(s, ω) + k2U(s, ω) = 0

r U(s, ω) r♣rs♥ts tr t tr ♦r ♠♥t k s t ♣r♦♣t♦♥♥r♦♥♠♥t ♥♠r ♥ rtt♥

k = ω√ǫµ =

2π

λ=

2π

λ0

√ǫrµr

t λ ♥ λ0 t ♥t ♥ t ♣r♦♣t♦♥ ♥r♦♥♠♥t ♥ ♥ t ♠rs♣t②

ǫr ♥ µr t rt ♥r♦♥♠♥t ♣r♠tt② ♥ ♣r♠t② ♥ t

ǫr = ǫ′r − j60σλ

µ = µ′ − jµ′′r

σ t ♥r♦♥♠♥t ♦♥tt②❯s♥ ♦t t ① qt♦♥s ♥ ♠♦t③ qt♦♥ t ♦♦♥ ♣r♦♣rts

♦♥ t tr♦♠♥t s ♥ rtt♥ ss♠♥ ♥ ♦♠♦♥♦s ♣r♦♣t♦♥ ♥r♦♥♠♥t t ♥r② ♣r♦♣t ♦♥ strt ♥ ♦rt♦♦♥ t♦ t r♦♥t ❲ r♦♥ts r ♣♥ s♣r ♦r②♥r srs r t ♠♣t ♥♦r ♣s r ♦♥st♥ts st ♦ r②s ♦r t s♥ st s♠♥ts ♥ ♦♥ r

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

♥r② rr ② r② s ♦♥t♥♦s ♥ ♦r ♣♦♥t ♦ t♠ ♥ s♣♦t ♥ ♠♣t ♥ ♣s

②s r ♦♠♣♥t t t r♠t ♣r♥♣ stss tt t ♣tt♥ t♥ t♦ ♣♦♥ts ② r② s t ♣t tt ♥ trrs ♥ t stt♠ ♥ ♦♠♦♥♦s ♥r♦♥♠♥t ts ♦rrs♣♦♥s t♦ t t tt tr♦♠♥t s ♣r♦♣t ♥ strt ♥

A

B

C

D P(0)P(s)

Caustics

ts ♥ ♦ ss

♦t② Pr♥♣

t ②s ①sts s r t ♥ ♦♥sr ♦② ♣♥ s ♣♣r♦①♠t♦♥ ♦s t♦ ♥r③ t ♣r♥♣ ♦ ♣♥ s t♦ tr♦♠♥t ss s ♥♦♥ s t ♣r♥♣ ♦ ♦t② ♦ tr♦♠♥t s ♦t tr ♥♠♥t s r ♣r♣♥r t♦ t ♣r♦♣t♦♥ rt♦♥ s s t tr t♦r ♦♥s t♦ ♣♥ s ♣r♣♥r t♦ t ♣r♦♣t♦♥ rt♦♥ trr♦♥ (E,H, s) s ♦rt♦♥♦r♠ rt ♥ s t♦

E =

√

µ

ǫ(H× s)

s t s ♣♦ss t♦ ♠t t st② t♦ t tr E ①♣rss ♥ t ♦ss B = (s, θ, φ) ♣♥♥t ♦ t ♣r♦♣t♦♥ rt♦♥

①♣rss♦♥ s♥ ♦♠tr ♣ts

♦♣♠♥t ♦ t tr ♥ ♦r♠s♠ rsts r♦♠ t ♦rs ♦♥r ♥ ♥ ♦♥ t ①s qt♦♥s

E(s) = A(s, ρ1, ρ2)E(0)e−jkψ(s)


♣♣r♦

E(s) =

0E‖(s)

E⊥(s)

B

= A(s, ρ1, ρ2)

0E‖(0)

E⊥(0)

B

e−jkψ(s)

r A(s, ρ1, ρ2) s t ♠♣t rt♦ t♥ E(s) ♥ E(0) s ♥♦♥ s t r♥ t♦r

A(s, ρ1, ρ2) =

√

ρ1ρ2(ρ1 + s)(ρ2 + s)

s s t st♥ t♥ P (0) ♥ P (s) r ψ(s) = s s ♣s ♥t♦♥ ♦r ♥ ♦srt♦♥ ♣♦♥t P (s) ρ1 ♥ ρ2 r t t♦ ♣r♥♣s rs ♦ rtr ♦ t r♦♥t ♠sr♦♥ t r② r♦♠ t rr♥ ♥tr ♣♦♥t P (0) s ♣♦♥t ♦rrs♣♦♥s t♦ t♣♦st♦♥ ♦ t s♦r ♦r t♦ ♥ ♥trt♦♥ ♣♦♥t t♥ t ♥ t sr

s ♦♦♥ t ♣♥ ss♠♣t♦♥ t tr s ss♠ ♦rt♦♦♥ t♦ t ♣r♦♣t♦♥ rt♦♥ s t ♦♠♣♦♥♥t ♦♥ t ①s s s q t♦ ③r♦ s t♥ ①♣rss ♥ ssB ♣♥♥ ♦♥② ♦♥ t ♣r ♥ ♣r♣♥r♦♠♣♦♥♥ts ♦rt♦♦♥ t♦ t rt♦♥ ♦ ♣r♦♣t♦♥

E(s) =

[

E‖(s)

E⊥(s)

]

B

= A(s, ρ1, ρ2)

[

E‖(0)

E⊥(0)

]

B

e−jks

t♦ rs ♦ rtr ρ1 ♥ ρ2 r ♠♦ ♦♥ t r② trt♦r②♥ t♠ t r② ♥♦♥tr ♥ ♦st ♦♥ ts trt♦r② t♦s rs ♦rtrs t♦ ♣t ♣♥♥ ♦♥ t ♦st ♥♦♥tr ② t r② ♦♥♦ t ♥trt♦♥ sr ♥ sst♦♥ ♥ ♦r

♥trt♦♥ ♦♥ts

♦t♦♥ ♦ t tr ♣♥s ♦♥ t t②♣ ♦ ♥trt♦♥s ♥♦♥tr r♥ ♣r♦♣t♦♥ s st♦♥ sr t t♥ ♦♥ts ♦ t tr ♣♥♥ ♦ t ♥trt♦♥ t②♣

t♦♥ ♦♥t

♥ r r ♣rs♥t rt♦♥ ♥ rrt♦♥ ♣♥♦♠♥ ♥tr♣♦♥t ♦♥ t tr ♠tr sr s Q ♥s θiθr ♥ θt s♥tt ♥♥ ♥ t rt♦♥ ♥ ♥ t rrt♦♥ ♥ rs♣t② ♦s♥s r ♥ ♥ rr ♦ t t♥♥t ♣♥ ♥♦r♠ n rt t♥ PR t st♥ sr ♦ t rt♦♥ ♣♦♥t Q ♥ rtt♥

[

Er‖(sr)

Er⊥(sr)

]

Br

=

√

ρr1ρr1 + sr

ρr2ρr2 + sr

[

R‖ 0

0 R⊥

] [

Ei‖(si)

Ei⊥(si)

]

Bi

e−jksr

♦♥t①t r ②r♥ ❯❲ ♥ Pr♦♣t♦♥

①♦♥ ♥ rrt♦♥ ss

r ρr1 ♥ ρr2 r t ♣r♥♣ rs ♦ rtr ♦ t rt r♦♥t t trr♥ ♣♦♥t ♥ Br = (sr, er‖, e

r⊥) ♦ rt rt s ♥ s t sr s

♣♥ ρr1 = ρi1 ♥ ρr2 = ρi2♦♥sr♥ tt t sr s ♥♠t ♥ s ♣r♠tt② ǫr t s ♣♦ss t♦

rt t s♦t rt♦♥ ♦♥t R‖ ♥ t r rt♦♥ ♦♥t R⊥ ♥ t sr θi = θr t

R‖ =ǫr cos θ

i −√

ǫr − sin2 θi

ǫr cos θi +√

ǫr − sin2 θi

R⊥ =cos θi −

√

ǫr − sin2 θi

cos θi +√

ǫr − sin2 θi

t ♠② ♥♦t tt ♥ s t sr s ♣rt② ♦♥t♥ t rt♦♥♦♥t ♦♥t ♣♥ ♦♥ t ♥t ♥②♠♦r ♥ r q t♦

R‖ = +1

R⊥ = −1

r♥s♠ss♦♥ ♦♥t

♠r② t tr♥s♠tt ♥ rtt♥

[

Et‖(st)

Et⊥(st)

]

Bt

=

√

ρt1ρt1 + st

ρt2ρt2 + st

[

T‖ 0

0 T⊥

] [

Ei‖(si)

Ei⊥(si)

]

Bi

e−jkst


♣♣r♦

r ρt1 ♥ ρt2 r t ♣r♥♣s rs ♦ rtr ♦ t r♦♥t tr♥s♠ttt♦ t rr♥ ♣♦♥t ♥ Bt = (st, et‖, e

t⊥) ♦ rt tr♥s♠tt s ♥ s

r t sr s ♣♥ ρr1 = ρi1 ♥ ρr2 = ρi2 t ♥tr s ♣♥ t ①♣rss♦♥

♦ t rs ♦ rtr ♥ rtt♥ ❬❪

ρt1 = ρi1α−1

ρt2 = ρi2α−1γ−2

t α =

1√ǫr

♥

γ =cos θi

cos θt

♥ s ♦ ♥♦♦r ♣r♦♣t♦♥ s r ♦♥sttt ♦ r♥t ②rs ♦ r♥t♠trs st s s ♥ t rt♦♥ ♥ rrt♦♥ ts rr♣t t ♥tr t ♥ ♠tr

s t tr♥s♠ss♦♥ ♣r♠tr s ♦♠♣① ♥t♦♥ ♣♥s ♦♥ t t②♣♥ t ♥♠r ♦ st ♠tr ♥t♦ t s t r♥t ♥tr ♦♥sttt♥t s rt♦♥ rrt♦♥ ♣♥♦♠♥ ♣♣rs ♦r♥ t♦ ❬❪ t s ♣♦ss t♦rt t ♦♠♣♦♥♥ts ♦ t ♦t♦♥ tr s s♠ ♦ t tr s ♦t♦♥r♦♠ ♠trs s ♣rtr ♣♦♥t s trt ♥ st♦♥

②♦ts r♣s sr♣t♦♥

♣r♦s st♦♥ s ♣rs♥t stt ♦ t rt ♦♥ ♣r♦♣t♦♥ t♦♦s♥ s♦♠♦♥srt♦♥s ♦♥ t ♣r♦♣t♦♥ ♥♥ ♠♦s s♣② ♦♥ t ❯❲ t♥♦♦②❲t ts st♦♥ ♥s t sr♣t♦♥ ♦ t t♦♦ ♦r ♦③t♦♥ ♥♣rtr ts st♦♥ ♦ss ♦♥ t r♣ s ♥r♦♥♠♥t sr♣t♦♥

r♣s ♦tt♦♥

r♦♠ t r♣ t♦r② r♣ s ♦♥♥t st ♦ ♥♦s r♣s ♥ ♦r♥t♦r ♥♦♥ ♦r♥t ♥ t sq rr t♦ r♣ s♦ s s♣ t strtr ♥ tt s t♦s r♣ t strtrs ♠ s♣ t♦♥rsss♦t t ♥♦s ♦r s t♦ st♦r t r♥t ♣ ♦ ♥♦r♠t♦♥ rt t♦ tr♥tr

r♦♦t t ♠♥sr♣t t ♥tr♦ r♣s ♥① t ttr xrt t♦ tr ♥♠

♦♦♥ ♥♦tt♦♥s ♦r r♣ sr♣t♦♥ r s r♣ ♥① x r♦♠ ts ♥♠ s ♥♦t Gx(Vx,Ex) r Vx ♥ Ex r rs♣

t② t st ♦ ♥♦s ♥ t st ♦ s ♦ t r♣ Gx♦r r♣ Gx(Vx,Ex) ♥ N ♥♦s ♥ E s

Vx = v0x , . . . , v

N−1x


Ex = e0x, . . . , eE−1x

r ♦r ♥ ♦r♥t r♣

ekx = (t(ekx), (ekx)) ∈ Vx × Vx

r♣s ♦r ②♦t sr♣t♦♥

s st♦♥ ♥tr♦ t r♣s rqr ♦r ②♦t sr♣t♦♥

t r♣ ♦ strtr Gs(Vs,Es) ♥♦♥ ♦r♥t t t♦♣♦♦ r♣ Gt(Vt,Et) ♥♦♥ ♦r♥t

♦r s♥trs tr♠♥t♦♥ t r♣ ♦ st② Gv(Vs,Ev) ♥♦♥ ♦r♥t t r♣ ♦ ♥trt♦♥s Gi(Vi,Ei) ♦r♥t

Gs(Vs,Es) t r♣ ♦ strtr

r♣ Gs s s t♦ sr t s♠t♦♥ ♥r♦♥♠♥t t s t rst r♣t♦ ♥ s r♣ s rt② ♦t♥ r♦♠ ♥♦ st ♥ st s r♣s ♥ ♠♦ ♦♥ t ②♦t t♥ ♣s s ♦♥ t tt♥ t♦r t ♥ s♦ ♥rt r♦♠ ♣r♦♣r ♦♥rs♦♥ r♦♠ ♦tr ①st♥♦r♠t s ❳ ♦♣♥ strt ♠♣ ♦r♠t ♦s♠ ❬❪ t ♦tr r♣s r rr♦♠ ♣r♦ss♥ ♦♥ ts ♥♦s ♥ s

♥ t ♥♦♦r s ②♦t s ♥r② ♥ ♦♦r ♣♥ s♠♥t sr♥ t ②♦t ♦t♥s s ss♦t t♦ ♥♦ ♦ Gs s t t♦ ♣♦♥ts ♦♥♥ s♠♥ts r s♦ ss♦t t♦ t♦ ♥♦s ♦ Gs ♦♥♥t♦♥② ♥♦s ♦ Gs ss♦t t♦ s♠♥ts r ♥♦t t ♣♦st ♥tr rs ♥♦s ss♦t t♦♣♦♥ts r ♥♦t t ♥t ♥tr ② ♦♥strt♦♥ ♥♦s t ♣♦st♥① s♠♥ts r ♥ t♦ t♦ ♥♦s t ♥t ♥①s ♣♦♥ts ♦t tt t♥rs s ♥♦t tr ♣♦♥t ♥ ♦♥♥t t♦ ♠♦r tt t♦ s♠♥ts

r s♦s ♥ ①♠♣ ♦ t ♥rt r♣ ♦ strtr GsP♦st ♥♦s ♦ Gs tt ♠tt t♦♥r② ♦♥t♥♥ t ♥♦r

♠t♦♥ st ♦ s s tr ♦r rtr ♠♥ ♦r tr♦♠♥t♣r♦ss♥

st ♦ t ♦♥♥t ♣♦♥ts ♥①s t ♥♦s ♦ Gs t ♥t♥①

t ♥♠ ♦ t s ss♦t t♦ t s♠♥t ♣♦st♦♥ t ♥♦r♠ t♦r ss♦t t♦ s♠♥t ♦♦♥ tr♥st♦♥ ♥t♦r st ♦ ♦♥♥t ♣♦♥ts s rt② r♦♠ t r♣ ♦♥♥tt②

ss ♥♠ s s♣ r♥ t ②♦t t♦♥


♣♣r♦

①♠♣ ♦ r♣ ♦ strtr Gs ❲t ♦ts r♣rs♥t s♠♥ts ♥♦s t♣♦st ♥① ♥ t ♦ts r♣rs♥ts ♣♦♥ts ♥♦s t ♥t ♥① ♦ t②♦t

s s st ♦ r♥t ♦♥sttt ♠trs ♦ r♥t t♥ss ssr♣t♦♥ rqrs t ♥♦ ♦ s ♠♥ts t tr tr♦♠♥t ♣r♦♣rts s sr♣t♦♥ s ♦♣♣ ♥ st♦♥

♣♦st♦♥ s ♥ s t♦r ♥ R3 t ♥ t♦♥ s ♦ t♦

rss stt♦♥s r s♠♥t s ♠ t r♥t ss t r♥t ts st ♦ ② ♥♠rs ♦♥t♥s ②s t ♠♦st ♥♦ ♥ srr♦♥ ② t♦②s s ♥♦r♠t♦♥ s ♦♥ t t♦♣♦♦ r♣ Gt s ♥ t ♦r♥♦r♠t♦♥ ♦t Gt ♥ ②s s ♣r♦ ♥ s♥ ♥♦r♠ t♦r ss t♦ tr♠♥ t ♦r♥tt♦♥ ♦ t s♠♥t tr♥st♦♥ ♥t♦r ♥ts t s♠♥t ♥ ♠♥② r♦ss ② ♥ ♥t

♥ ts ♦♥♥t♦♥ tr♥st♦♥ ♥t♦r s r② s ♦r sr♥ r s rs r rt ♣rtt♦♥ t ♥tr ♠♥ ♥♦r tr♦♠♥t ♦♠♣tt♦♥ t strt♦♥ s ♥ ♥tr♦ t♦ r ♦♠♣①t② r♥ rr♣ ♦♠♣tt♦♥

r♣ ♦ ♦♣♦♦② Gt t♦♣♦♦ r♣ Gt ♠s t♦ strt t strtr r♣ Gs t♦ ②s r♣r

s♥tt♦♥ r♣ ② s ♦s ♣t strt♥ ♥ ♥s♥ t t s♠ ♥♦ r ♥♦s

♥ s ♦ tt ♣t ♥ ♦♥② s ♦♥②s r ♦♥♥♥t t♦ ♦t♥ strtr ♥♦r♠t♦♥ ♦ t ②♦t ♥

♦rr t♦ ♥rs t ♣r♦ss♥ s♣ ♥ r ♦tr s r♣s ♦r t ♣r♦♠t ♥ Gt s t r♦♠ Gs ♥♦s ♥ rs♣t ♦ tr rs

②s t♦ s♦♥t ♠♥s ♥♦♥ ② ♠st ♥ ♥t♦♥♦tr

♥♦s ♦ Gs t ♥t ♥① s t♦ s ♦r ♥ ②s ♦ t♦s ♥♦s ♥ ♦♥② s ♦r ♥ t♦ ②s t ♠♦st ♥♦ ♦ Gs t ♣♦st ♥① ♥ ♥ ②s t ♠♦st rst st♣ t♦ Gt s t♦ ♦t♥ ② ss r♦♠ t ♦♠♣♦st♦♥ ♦ Gs

s Gs s ♥ ♥rt r♣ t ♥ ♥ s ② s♣ t ①sts st ♦


♥r② ♥♣♥♥t ②s ♥♠ ss ②s ♥ r♣rs♥t Gs ❬❪ ss ② ♦♠♣♦st♦♥ s ♣r♦r♠ ② s♥ s♣ ♦rt♠ sr

♥ ❬❪ ♦rt♠ srs t ♥ ♣r♦ss ♦ Gt

♦rt♠ ♥ t♦♣♦♦ r♣ Gt

qr GsC = ②❴ss(Gs)Gt = r♣(C)Gt = r♠♦❴♥❴②s(Gt)Gt = ♦♠♣♦s(Gt)

♥t♦♥ ♥t ②s ♦ ②s r ♥t t② sr t st ♦♥s♠♥t ②s r ♦♥② ♦♥♥t tr♦ s♥ ♣♦♥t r ♥♦t ♦♥sr♥t ♥ t♦ r ♥♦t ♦♥♥t

s s♦♥ ♥ r ♥♦s ♦ t rst♥ Gt r♣ r♣rs♥t ②s ② ♦♥strt♦♥ s ♦♥♥t ♥t ②s

0

1 23456

7

8

910111213

14 15

16

♥ ①♠♣ ♦ t♦♣♦♦ r♣ Gt ♦s ♦ ts r♣s r ②s s♦ ts r♣ ♦♥♥t ♥t ②s

♦t Gs ♥ Gt r rqr ♦r ♣rs sr♣t♦♥ ♥ ♦r♥③t♦♥ ♦ ②♦t② r s♦ ♠♥t♦r② t♦ r r♣s rt t♦ ♣r♦♣t♦♥

r♣s ♦r ♣♥ tr♠♥t♦♥ ♦ ♥trs

♥ t♦♥ t♦ strtr ♥ t♦♣♦♦ ♥♦r♠t♦♥ ♣r♦♣t♦♥s♣ r♣♠st t t♦ ♣r♦r♠ ❯ s r②tr♥

r♣ ♦ ❱st② Gv

r♣ ♦ st② Gv s s t♦ sr t ♦♣t st② t♥ ♣♦♥ts♥ s♠♥ts ♦ ②♦t

♥♦s ♦ Gv r t s♠ s t♦s ♦ Gs s ♦ Gv ♦♥♥t ♥♦s rttr♦ ♥ ♦♣t st② rt♦♥s♣ ♦s st② rt♦♥s ♠t ♦♥r♥ t♦s♠♥ts ♣♦♥t ♥ s♠♥t ♦r t♦ ♣♦♥ts


♣♣r♦

♣♥♥ ♦♥ t ♦♠tr② s♦♠ ♣♦♥ts r ♥t ♦r ♦♠♥ rt♦♥♣♦♥ts s♠♣ r ♦r tr♠♥♥ rt♥ ♣♦♥t ①♣♦ts t s♥ ♥ tr ♦ t ♥♦

♥t♦♥ rt♥ ♦ rt♥ ♥♦ νs s s tt

(νs < 0) ♥ (r(νs) ≤ 2)

r r ♦ ♥♦ s♥t t ♥♠r ♦ s ♥♥t t♦ t rt①

♦rt♠ srs t r♥t st♣s ♦ ♥ Gv s s ♥♦♥qr ♣♣r♦ r♣ ♦ st② ♦r ② ♦ t ②♦t s ♦t♥♥♣♥♥t② ♥ ♦♠♣♦s ♥ ♦♦♣ t♦ ♦t♥ t ♦r r♣ ♥ r♥t s♥ ♠♣♠♥t r ♦ ♥♦t ♦♠♣♦s r♣ t ♣ r♣ ♦ st②s♣rt ♥ t♦♥r② ♥① ② t ② ♥♠r

♥ ①♠♣ ♦ t rst♥ r♣ ♣r♦ss ② ts ♦rt♠ s s♦♥ ♥ r

♦rt♠ r♣ ♦ st② Gv

qr Gs GtGv = r♣() ⊲ ♥st♥tt ♦ r♣♦r cycle ♥ Gt ♦

Gv(cycle) = (Gs, cycle) ⊲ Gv ♦r cycleGv = ♦♠♣♦s(Gv,Gv(cycle))

♥ ♦rtr♥(Gv)

1

2

3

4

5

6

7

8

9

10 11 12 13 14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42 43

44

45

46

47

48

49

50

51

52

53 54

55

5657

5859

6061

6263

6465

66 67

6869

7071

7273

7475

76 77

7879

8081

82 83

8485

8687

88 89

-71 -70

-69 -68

-67 -66

-65 -64

-63 -62

-61 -60

-59 -58

-57 -56

-54

-53 -52

-51 -50

-49 -48

-40-23

-22

-2

-9-8

-7-6

-5-4

-3 -1

r♣ ♦ ❱st② Gv

♥ ♠♣♦rt♥t rqr♠♥t ♦r tr ♦♣♠♥t ♥ s t♦ rtr t t②♣ ♦♥trt♦♥ r ♦♥ t② s t♦ ♠ t st♥t♦♥ t♥ rt♦♥ ♥ tr♥s♠ss♦♥ ♣r♦♣♦s ♠♥♥r t♦ s♦ ts ♣r♦♠ s t♦ ♥tr♦ ♥ r♣ rr♦♠ Gv ♠s r ♦r s♠♥t t s t ♥♦ t②♣ ♦ ♥trt♦♥


r♣ ♦ ♥trt♦♥s Gi

r♣ ♦ ♥trt♦♥s Gi s r r♦♠ t Gv r♣ ♦ ♥trt♦♥s Gi s rt r♣ trs t ♣♦ss ♥trt♦♥s t♥ ♥♦s Vs ♦ ♥② ♦ Gt

Prt② s♥ ♥trt♦♥ ♦♥ s♠♥t ♦ Gv ♥ ♣♦t♥t② ♥rt ♣ t♦4 ♥trt♦♥s ♥ Gi rt♦♥ r♦♠ s ♦ t ♥trt♦♥ ♥♦r t rt♦♥r♦♠ t ♦tr s ♥♦r t tr♥s♠ss♦♥ ♥ ♦♥ ② ♥♦r t tr♥s♠ss♦♥ ♥ t♦tr ② r s♦s ts st♥t♦♥s

(4,0,1)

(4,1,0)

0 1(4,1)

(4,0)

4 ♣♦t♥t t②♣s ♦ ♥trt♦♥s ♦♥ ♥ ♠tr

r t②♣s ♦ tr♦♠♥t ♥trt♦♥s r t♦ ♦♥sr t tr♥s♠ss♦♥ t rt♦♥ ♥ t rt♦♥ ♦r♥ t♦ t ♥♦tt♦♥ ♥ ♥

(v is, vkt ) s rt♦♥ ♦♥ s♠♥t v is > 0 ♥ ② vkt

(v is, vkt , v

lt) s tr♥s♠ss♦♥ r♦ss s♠♥t v is > 0 ♥ r♦♠ ②s vkt t♦ ② v lt

(v js ) s rt♦♥ ♦♥ ♥♦ v js < 0

♦rt♠ srs ♦ s rt Gi r♦♠ Gv rst♥ r♣ s s♦♥ ♥r

♦s ♦ Gi ♦♥ssts ♦ tr♦♥♦s ♥trt♦♥s t♣s ♥ rtt♥ t ♥ ♦sr tt t ♥♦ ♥♠s ♦ Gi r t♣s ♦♥t♥ ♥♦r♠t♦♥ ♦t

t ♥① ♦ ♥♦ s♠♥ts ♦r ♣♦♥ts r♦♠ t Gs ♥♦s t②♣ ♦ ♥trt♦♥ r①♦♥ ♦r rt♦♥ ♣♥♥ ♦♥ t ♥♠r ♦ ♠♥ts♥ t♣

t s♣ sr♣t♦♥ ♦ r② ♦♥♥♥t ♥ t ♥ st♦♥ ♦r t ♥tr sr♣t♦♥


♣♣r♦

♦rt♠ r♣ ♦ ♥trt♦♥s Gi

qr Gs Gt Gv⊲ Gi ♥♦s♦r n ♥ ♥♦s(Gv) ♦ ⊲ ♦♦♣ ♦♥ Gv ♥♦s

n < 0 t♥ ⊲ ♥♦ s ♣♦♥tGi❴♥♦((n)) ⊲ rt♦♥ ♥♦

s ⊲ ♥♦ s s♠♥t②0②1t❴②s(Gs, n)Gi❴♥♦s((n, cy0, cy1), (n, cy1, cy0)) ⊲ tr♥s♠ss♦♥sGi❴♥♦s((n, cy0), (n, cy0))) ⊲ rt♦♥s

♥ ♥ ♦r

⊲ Gi s♦r inter1 ♥ nodes(Gi) ♦

linter = s♥tr(Gv, inter1) ⊲ ♥ ♥♦s ♦ Gi ♥ st② r♦♠ ♥♦s ♦ Gv♦r inter2♥ linter ♦

inter2 s t♦♥ ♦r inter2 s rt♦♥ t♥Gi❴ ((inter1, inter2))

s inter2 s r♥s♠ss♦♥ ♥ inter1s r♥s♠ss♦♥ t♥Gi.add❴edge((inter1, inter2))

♥ ♥ ♦r

♥ ♦r


-5

1

2

3

4

5

6

7

0 1

-68

9

10

11

12

-7

-8

2

-1 -2

-3-4

-10

-9

(7,1)

(8,1)

(9,2)(9,1)(9,1,2)

(9,2,1)(-8)

(-7) (10,2)

(11,2)

(12,2)

(5,1)

(6,1)

(4,1)

(4,0)

(4,0,1)

(4,1,0)

(2,0)

(1,0)

(3,0)

r♣ ♦♥ t t♦♣ ♦♠♥s sr r♣s s♠♥ts ♥♦s ♣♦st♥♠r ♥ t ♦ts ♥ ♣♦♥ts ♥♦s ♥t ♥♠rs ♥ r② ♦ts r r♦♠ Gs♥ t ② ♥♠r ♥♠r ♥ r sqr r r♦♠ Gt r♣ ♦ s t ss♦t r♣ ♦ ♥trt♦♥ Gi ♥♦s ♦rrs♣♦♥♥ t♦ rt♦♥ tr♥s♠ss♦♥ ♥rt♦♥ r ♦♦r ♥ t ♦r♥ ♥ r♥ rs♣t② ♠♥t 4 ♦rrs♣♦♥st♦ r♥t t②♣ ♦ ♥trt♦♥s (4, 0) rt♦♥ t♦r ② 0 (4, 1) rt♦♥ t♦r② 1 (4, 0, 1) tr♥s♠ss♦♥ r♦♠ ② 0 t♦ 1 ♥ (4, 1, 0) tr♥s♠ss♦♥ r♦♠ ② 1t♦ 0


♣♣r♦

r♦♠ r♣s t♦ ② ♥trs

♥ t ♦♦♥ s ♥tr♦ t ♥ ♦♥♣t ♦ r② s♥tr rst t ♠♦tt♦♥s♥ tt ♦♥♣t ♥ t ♥t♦♥ ♦ s♥trs r ♥ rtr t ♠t♦t♦ ♦t♥ t♦s s♥trs r♦♠ t ♣r♦s r♣ sr♣t♦♥ s t

♦tt♦♥ ♦ t ♥tr♦t♦♥ ♦ ② ♥tr

♣r♦♣t♦♥ ♥♥ ♣r♠trs r t♠ r②♥ t♦ t ♠♦t♦♥ ♦ t ①tr♠ts ♦ t r♦ ♥ ♦r t ♠♦t♦♥ ♦ ♠♥s ♦r ♦ts ♦♥ ♥ t ♣r♦♣t♦♥♥r♦♥♠♥t ♥ s♣t ♦ t♦s ♠♦♠♥ts ♣t ♣rsst♥② s ♥ ♥ ♥ ❬❪ st ♦t♦♥ ♦ ♣rtr ♣t ♦ t ①ts r♥t ♥ ♦ ♥ tr ♥ ♦r♥ t ts r♥t ♠♦t♦♥ ♦♥sr♥ ♣♣r♦s r♥t ♥ ♥t♦ t s♦ rt t♦ s♦♠t♥ ♦sr ♥t♦t r②s ♦r ♠♦r ♣rs② ♥t♦ tr rt sts ♦ ♥trt♦♥ ♣♦♥ts t ♣♣rs ttt② r rt ② t ♥tr ♦ t ♥♦ ss ♥ t t②♣ ♦ ♥trt♦♥s ♦♥t♦s ss

s ♦♥ ts ♦srt♦♥ t ♥ ♥ r②s♥tr ♦r s♥tr s ♥♥ ♦♥trt♦r ♦ t stt♦♥r② ♣rt ♦ t ② ①t♥s♦♥ t stt♦♥r②♣rt ♦ t s ♦t♥ ② rt♥ t s♥trs

♥ ♥ t ♥ ♦sr tt s♥tr s ♥♦t r② rt t♦ tr♥s♠ttr ♦r rr ♣♦st♦♥ ♥ ♥ t s♠ ♦r s♠ ♦r ♥ s♦♠ ss r♣♦st♦♥ ♥ s ♠♥s tt t ♥ s ♦r ♥rt♥ ♦ st ♦ r②st♥ t♦ r♥t r♦♥s s ♦ r♣ ♦r ♠♦♥ st♦st ♥♥ ss♦ ♥ ♥tr♦ ♥ ♦♣ ♥ ❬❪ t♦ sr t ♥♥ rrrt♦♥ ♥s r♦♦♠ s st② s s♦♥ tt t rrrt♥ ♣rt ♦ t ♥s rrrt♥ ♦♠ ♥ ①♣♥ ♥ r♣r♦ t t② ② s♥ r♣ sr♣t♦♥ ♥s♣r ② ts r♣rs♥tt♦♥ t s ♣r♦♣♦s ♥ t ♦♦♥ t♦t ♥t ♦ r♣ sr♣t♦♥ ♦r ♥ st s♣ s♥trs

t♥♥ t ♥trs r♦♠ Gi

♦♥♣t ♦ s♥tr s ♥tr♦ t♦ r②s r② r s ♥ ♥ s ♦t♥ ♥ (O, x, y) ♥ t t ♥♦tt♦♥ ♥ts tt tr ♦♠♣♦♥♥tsr ♥ t ♦♥② ♠♥s♦♥s ♥ r② r s sq♥ ♦ ♥trt♦♥s ♣♦♥tspl ♥ ♥ (O, x, y)

♥t♦♥ r②s r②s r ♣♦②♥s ♦s s♠♥t s ♠t ②♥trt♦♥ ♦r tr♠♥t♦♥ ♣♦♥ts ♥ (O, x, y) r② r t Lh ♥trt♦♥ ♣♦♥ts s♥ ♦rr st ♦ Lh+2 ♣♦♥ts ss♠♥ tt ♣♦♥ts p−1 ♥ pLh

r♣rs♥t♥ trt tr♥s♠ttr ♦r t rr r ♥ rst ♥ st ♣♦st♦♥ rs♣t② s r② ♥ ♥ rtt♥

r = [p−1, p0, . . . , pLh−1, pLh]

tt♥ ②s ♦t ♦ ♥trs

s ♥ ♥ s♥tr s t ss♦t♦♥ ♦ ♦rr sq♥s t ♥tr♦ t s♣♣♦rt s ♥ t t②♣ ♦ t ♥trt♦♥s ♥♦s ♦ Gi s sr♥ st♦♥ ♦♥t♥ ♥♦r♠t♦♥ ♦t t ②s ♥♦ ♥ t ♥trt♦♥ t♥tr ♦ t ♥trt♦♥ t ss♦t ♥① ♦rrs♣♦♥♥ t♦ s ♦s t♦ st♥♦r♠t♦♥ ♥ ♥t♦s② s t♦ s♥tr

♥t♦♥ ② ♥tr r② s♥tr s = S(r) ♦ r② r t Lh ♥trt♦♥s s t ss♦t♦♥ ♦ t♦ ♦rr sq♥ ♦ ♥t Lh rst sq♥ ♣r♦st ♥tr ♦ t s ♥ s♦♥ sq♥ ♣r♦s t ss♦t t②♣ ♦ ♥trt♦♥♠♦♥ ♥trs ♥ ♦t♥ tr r♦♠ tr♠♥t♦♥ ♦ ♣ts ♥ t♦r♥t r♣ Gi ♦r r r♦♠ ♥ ①st♥ r② s s♥tr ♥ rtt♥

s =

[

n0 . . . nl . . . nLh−1

t0 . . . tl . . . tLh−1

]

t♥♥ s♥tr r♦♠ sq♥ ♦ ♥♦s ♦ Gi s st♣s ♣r♦ss

tr♠♥♥ t♦ ssts ♦ ♥♦s Vp−1

i ♥ VpLh

i r♦♠ Gi r ♥ rtst② ♦ p−1 ♥ pLh

rs♣t② ♥♥ s♠♣ ♣ts ♣ts t ♥♦ r♣t ♥♦s t♥ ♥② st♥t ♣r♦ ♥trt♦♥s (V p−1

i ,V pL

i )♦s rtr♥ ♣ts r ♦rr sq♥s ♦ ♥♦s Gi ♥ s② tr♥s

♦r♠ ♥t♦ s♥trs ♦rt♠ srs tt ♣r♦r

♦rt♠ tr♠♥t♦♥ ♦ s♥trs

qr p−1pLh

qr GiS = ❬ ❪ ⊲ ♥t③ ♥ ♠♣t② stV

p−1

i ← ts♥trt♦♥s(Gi, p−1)

VpLh

i ← ts♥trt♦♥s(Gi, pLh)

♦r it in Vp−1

i ♦

♦r ir in VpLh

i ♦S.♣♣♥s♠♣♣ts(Gi, p−1, pLh

))♥ ♦r

♥ ♦rs = rs♣t♦s♥tr(S)


s st♦♥ srs t st♣s t♦ ♦t♥ r② r♦♠ s♥tr rst r② s ♦t♥ r♦♠ s♥tr ♦♥ t r② s ①t♥ t ♠t♣ ♦♦rrt♦♥s t♦ rt r②


♣♣r♦


tr♠♥t♦♥ ♦ r② ♥ t ♣♥ (O, x, y) r♦♠ ts s♥tr s st♣s♣r♦ss

rst st♣ ♦♥ssts ♥ ①trt♥ t t ♥ ♣♦st♦♥ ♦ ♥trt♦♥ s♠♥ts r♦♠ t s♥tr

s♦♥ st♣ ♦♥ssts ♥ t♥ ♥tr♠t ♣♦♥ts r♦♠ ♦♥ tr♠♥t♦♥ ♦r ♥ ♦♦♥ t sq♥ ♦ ♥trt♦♥s ♦ t s♥tr

tr st♣ ♦♥ssts ♥ t♥ ♥trt♦♥ ♣♦♥ts ♦♥ s♠♥ts tt ♣ ♦ ♣r♦s② ♦♥ ♥tr♠t ♣♦♥ts

Ps ①trt♦♥ ♦ ♥trt♦♥s P♦♥ts ♠♥ts

t ♦ t s♥tr ♦t♥ r♦♠ ♦rt♠ t s ♣♦ss t♦ t t s♠♥t♦r ♣♦♥t ♥♠r nl ♥ r♦♠ r♣ Gs t ss♦t tr♠♥t♦♥s (al ♥ bl) ♥t

al = [xal , yal ]T ,

bl = [xbl , ybl ]T ,

♥ s t ♥trt♦♥ s rt♦♥ t ♥ ♣♦♥ts ♦r s♣ ♥♦r♠t② ♣♦♥t s r♣rs♥t s s♠♥t ♦ ♥t t t t♦ tr♠♥t♦♥s ♥ ts♠ ♦♦r♥ts gl = (al,al)

♥ r♦♠ t ♦ s♥tr s ♦rr sts ♦ t ♥ ♣♦♥t ♦♦r♥tspa ♥ pb r ①trt

pa = [a0, ...,aLh−1]

pb = [b0, ...,bLh−1]

Ps tr♠♥t♦♥ ♦ q♥ ♦ ♥tr♠t P♦♥ts

sq♥ ♦ ♥tr♠t ♣♦♥ts s tr♠♥ r♦♠ t ♥♦ ♦ t tr♥s♠ttr ♦r rr ♣♦st♦♥ p−1 = [x−1, y−1]

T t sq♥ ♦ ♥trt♦♥s s♠♥ts♦t♥ ♥ ♣s ♥ s♦ r② ♠♣♦rt♥t② t t②♣ ♦ ♥trt♦♥ s t ♦♠♣tt♦♥ ♦ ♥tr♠t ♣♦♥ts ♣♥s ♦♥ t ♥tr ♦ t ♥trt♦♥ s ♦♦

t ♠ ♦ t rr♥t ♣♦♥t rt t s♠♥t t rr♥t ♣♦♥t ts t rt♦♥ ♥trt♦♥ ♣♦♥t ts sq♥ ♦ ♥tr♠t ♣♦♥ts s rtt♥ s (2× Lh) t♦r

m = [m0, ...,mLh−1]

t ml = [xl, yl]

T

m0 ♣♦st♦♥ ♣♥s ♦♥ t t②♣ ♦ t rst ♥trt♦♥ s


m0 = K0p−1 + v0

m0 = p−1

m0 = a0♥ ♥r② mk ♣♦st♦♥ ♣♥s ♦♥ t t②♣ ♦ t lth ♥trt♦♥ ml = Klml−1 + vl ml = ml−1

ml = alr Kl s rt♦♥ ♠tr① ♥ s

Kl =

[

al −bl−bl −al

]

♥ vl s tr♥st♦♥ t♦r ♥ s

vl = [cl, dl]T

t

al =(xal−xbl )

2−(yal−ybl )2

(xal−xbl )2+(yal−ybl )

2

bl =2(xbl−xal )(yal−ybl )


2

cl = 2xal (yal−ybl )

2+yal (xbl−xal )(yal−ybl )


2

dl = 2xal (yal−ybl )(xbl−xal )+yal (xbl−xal )

2


2

♦♦r♥ts ♦ ♥tr♠ts ♣♦♥ts r ♦t♥ ② s♦♥ t ♦♦♥ ♥rs②st♠

Am = y

A =

I2−K1| − I2|O2 I2

−KLh−1| − I2|O2 I2

y =

K0p−1 + v0|p−1|a0v1|z2|d1

vLh−1|z2|dLh−1

r I2 ♥ O2 r rs♣t② t ♥tt② ♠tr① ♥ t ♥ ♠tr① ♦ ♠♥s♦♥ z2 s 2 × 1 ③r♦ ♦♠♥ t♦r ♥♦tt♦♥ R|T |D st♥s ♦r t ♣♦ss♠tr① ♥trs ♣♥♥ ♦♥ t ♥trt♦♥ t②♣

r s♦s ♥ ①♠♣ ♦ t♦ ♥tr♠t ♣♦♥ts r♦♠ s♥tr ♥♦♥ rt♦♥s


♣♣r♦

tr♠♥t♦♥ ♦ ♥tr♠t ♣♦♥ts m0 ♥ m1 r♦♠ t r♥s♠ttr p−1

♣♦st♦♥ m0 s t ♠ ♦ p−1 tr♦ g0 t s♠♥t ♠t ② ♣♦♥t a0 ♥b0 m1 s t ♠ ♦ m0 tr♦ g1 t s♠♥t ♠t ② ♣♦♥t a1 ♥ b1

Ps ♦♠♣t♥ t ♥trt♦♥ P♦♥ts r♦♠ P♦st♦♥s ♦ ♥tr♠t P♦♥ts

t s ♥ t lth ♥trt♦♥ ♣♦♥t ♦♥ r② s pl = [x′

l, y′

l ]T s

♣♦♥t s ♦t♥ r♦♠ t ♥trst♦♥ ♦ s♠♥ts t ♥trt♦♥ s♠♥t sl ♥t r② s♠♥t ♦♥♥ ml ♥ ml−1 ♥ s t t♦ s♠♥ts ♦ ♥♦t ♥trst♥ ♦tr t r② ♦ rt

♥ ♦rr t♦ tst t ①st♥ ♦ ts ♥trst♦♥ ♣r♠tr③t♦♥ t♦r Γl =[αl, βl]

T s ♥tr♦ ♥ t ♥♥♦♥ ♦ t qt♦♥ s②st♠ ♦♠s (4 × 1)

♦♠♥ t♦r s x′

l =[

pTl ,ΓTl

]T

♥trt♦♥ ♣♦♥ts r ♦♥ strt♥ r♦♠ t rr pLh= [xLh

, yLh]T r♦♠

t rr ♥ ♦♥♥ t st ♥tr♠t ♣♦♥t mLh−1 t ♥ rtt♥

p0 + α0(pLh−mLh−1) = pLh

p0 + β0(aLh−1 − bLh−1) = aLh−1

♣♦st♦♥ ♦ t rst ♥trt♦♥ ♣♦♥t s ♦♥ ② s♦♥ ts ♥r s②st♠

W0x′

0 = y′

0

t

W0 =

[

I2 pLh−mLh−1 02×1

I2 02×1 aLh−1 − bLh−1

]

y′

0 =[

pTLh,aTLh−1

]T

♥①t ♥trt♦♥ ♣♦♥t pl s tr♠♥ ♥t② t♥ ♥t ♦ t♥♦ ♦ ♣♦♥t pl−1


pl + αl(pl−1 −mLh−(l+1)) = pl−1

pl + βl(aLh−(l+1) − bLh−(l+1)) = aLh−(l+1)

Wlx′

l = y′

l

t

Wl =

[

I2 pl−1 −mLh−(l+1) 02×1

I2 02×1 aLh−(l+1) − bLh−(l+1)

]

y′

l =[

pTl−1,aTLh−(l+1)

]T

♣r♠tr③t♦♥ ♦ t s♥tr s♠♥t ♣②s r② ♠♣♦rt♥t r♦ 0 <βl, αl < 1 t ♥trt♦♥ ♣♦♥t pl s ♥ t t♦♥ ♥ ♣r♦ ♦trs t♥trt♦♥ ♣♦♥t ♦s ♥♦t ♥trst t s♠♥t ♥ t s♥tr s rt ♦r trr♥t rr ♥ t ♦♠♣tt♦♥ ♣r♦ss s ♥ ♦♥ s strt ♦♥ r t ♥trt♦♥ ♣♦♥ts pl r tr♠♥ ♠♥s tt t r s ♥♦♥

tr♠♥t♦♥ ♦ ♥trt♦♥ ♣♦♥ts p0 ♥ p1 p0 s t ♥trst♦♥ ♣♦♥t

♦ t s♠♥t [r,m1] ♥ g1 p1 s t ♥trst♦♥ ♣♦♥t ♦ t s♠♥t[

p‖0,m0

]

♥

s0 β0 ♥ β1 r ♥t ♣r♠trs ♦♥ g0 ♥ g1 rs♣t② α0 ♥ α1 r ♥t♣r♠trs ♦♥ s♠♥ts [r,m1] ♥ [p0,m0] rs♣t②

r♥ ②s r♦♠ ②s

tr♥s♦r♠t♦♥ ♦ r②s ♥ r②s s s ♦♥ ♥ t♦♥ ♦ r②s ts♦♥sr t r② r♣rs♥t ♥ r♦♠ t ♣r♦s ♦♠♣tt♦♥ t ♦t♥ r② s ♣r♦t♦♥ ♥ (O, x, y) ♦ t r② s s♦♥ ♥ r s t ♥ ♦sr r② s ♦♠♣♦s ♦

t ♦♠♣t ♥trt♦♥ ♣♦♥ts r♦♠ t r②s r t z ♦♠♣♦♥♥t s♠ss♥


♣♣r♦

③③t♦♥

♣r♦t♦♥ ♥ (O, x, y) t♦♥ ♦ ♥trt♦♥s ♣♦♥ts ♦♥t r② ①s s

♣rs♥tt♦♥s ♥ r♥t ①s ♦ s♠ r②

♥ ♥trt♦♥ ♣♦♥ts t♦ ♥♦r ♦♦r rt♦♥ r x ♥ y ♦♠♣♦♥♥tsr ♠ss♥ ss♠♥ t t♦♥ ♦ t s ♥♦♥

tr♥s♦r♠t♦♥ ♦♥ssts ♥ st♣s tr♠♥♥ t rt ♣♦st♦♥s ♦ ♦r ♦♦r ♥trt♦♥s ♦♥ t r② tr♠♥♥ t ♣♦st♦♥ ♦ t ♥trt♦♥s ♣♦♥ts ♥♥ t ♥ ♦♠♥

♦♦r ♥trt♦♥s ♥ (O, x, y, z) rt ♣♦st♦♥ ♦ ♦♦r ♥trt♦♥s ss r♣rs♥tt♦♥ ♣rs♥t ♥

r t r♣rs♥tt♦♥ s♦s t t♦♥ ♦ ♥trt♦♥ ♣♦♥ts s ♥t♦♥♦ t ♥t ♦ t r② ♥ t r t ♣♦②♥ r♣rs♥t♥ t r② ♥ (O, x, y) s♥ ♥r♣♣ ♥ s s t ♦r③♦♥t ①s ts ♥♦t t s ①s ♣♦st♦♥s♦ t ♥trt♦♥s r ts rt t♦ tr ♣♦st♦♥s ♦♥ t r② ♥tr♦t♦♥ ♦t♦s r♣rs♥tt♦♥s s rqr t♦ tr♠♥ t ♦♦r ♥trt♦♥s ♥ sst♦♥ ♥ s s t♦ tr♠♥ t ♣♦st♦♥s ♦ ♥trt♦♥s ♥ st♦♥

♥♥ t t P♦st♦♥s ♦ ♦♦r ♥trt♦♥s P♦♥ts

sr ♦ rt ♣♦st♦♥ ♦ ♥trt♦♥ ♣♦♥ts ♦♥ssts ♥ ♥♥ r t r②srt ♦♥ t ♦♦r ♥♦r t s ♠♥t♦♥ ♣r♦s② t sr ♦ t♦s rt♣♦st♦♥s s ♣r♦r♠ ♥ r♠ (O, z, s) t ♥♦ ♣♥ ♦♥ t r② trt♦r②♦r t s ♣♦ss t♦ rt ♦t t tr♥s♠ttr ♥ rr ♣♦st♦♥s ♥ tt ♣♥♥♦t a ♥ b rs♣t② ♥ t

a = [0, z−1]T

b = [(s, zL]T


♥♥ t ♥trst♦♥ t♥ t rt♦♥ ♥s Dm ♥ s♠♥ts r♦♠t ♠s ♦ a ♥ b ♦r Le = 2


♣♣r♦

r s ♥♦ts t ♥t ♦ t r② ♥ rtt♥

s =l=L∑

l=0

||pl − pl−1||

image ♠t♦ s s t♦ ♥ t♦s rt♦♥s s ♠t♦ ♦♥ssts ♥ ♥♥♥ ♦rr st ♦ Le rt ♣♦st♦♥ ♦ ♥trt♦♥s ♣♦♥ts q ♦rt② tt ♥♠r♦ sss rt♦♥s ♦♥ t ♦♦r ♥ ♦♥ t Le ♥ ♥♥t ♥ ♣rt tss ♣r♠tr ♦ t s♠t♦♥

♦r tt ♣r♣♦s ts ♥ t♦ ♥s D−1 ♥ D1 r♣rs♥t♥ t ♦♦r ♥ t rs♣t② ② ♥♦t♥ Dm−2|Dm−1 t rt♦♥ ♦ Dm−2 ② Dm−1 image lines Dm ♥ t

Dm =

Dm−2|Dm−1 for m ∈]2, Le]Dm+2|Dm+1 for m ∈ [−Le, 2[Dm−3|Dm−1 for m = 2

Dm+3|Dm+1 for m = −2

♠r② t s ♣♦ss t♦ ♥ t images points ♦ a tr♦ rt♦♥ ♥s Dm

t

am′ =

am′−1|Dm

′ if m′ ∈]0, ..., Le]

am′+1|Dm′+1 if m′ ∈ [−Le, ..., 0[

r am|Dm ♥♦t t rt♦♥ ♦ am tr♦ t ♠ ♥ Dms strt ♦♥ r t intersection points cm′

,m ♥ s t♥trst♦♥ t♥ s♠♥t ♠t ② am′ ♥ b ♥ rt♦♥ ♥ Dm

cm′ ,m = [am′ ,b] ∩Dm

♦rt♦♦♥ ♣r♦t♦♥ P s ♣♣ ♦♥ t ♥trst♦♥ ♣♦♥ts tr ♦♥ D−1 ♦rD1 ♣♥♥ ♦♥ t s♥ ♦ m

′

c⊥m

′,m

=

P (cm′ ,m,D1) if 2m′

+ 1;m′ ∈ N |k ∈ 2Z−

P (cm′,m,D−1) if 2m′

+ 1;m′ ∈ Z

− |k ∈ 2N

♦t tt ♣♦♥ts r② ♦♥ D1 ♦r D−1 ♦♥ t tr ♣r♦t♦♥s ♣t ♥ r t♦s ♦rt♦♦♥② ♣r♦t ♣♦♥ts r rt ♥tr

t♦♥s ♣♦♥ts ♣♦st♦♥s ♣ t♦ ♥ ♦rr♥ ♣r♦ss ss♠♥ ②s t s♠ ♦♥♥t♦♥r t t srs st ♦ t♦rs ♦ ♦♠♣♦♥♥ts t s ♣♦ss t♦ rt t♦rr sq♥ qm′ ,c ♦ ♥trt♦♥s ♣♦♥ts ♥ (O, z, s) t

qm′,c = [a, c⊥

m′ ,c, c⊥m′ ,2c

..., c⊥m′ ,(m′−1)c

,b]

r m′

s t ♥♠r ♦ s rt♦♥ ♥ c = −1, 1 t♦ ♦♦s st ♥ t ♥trt♦♥ ♦r ♦♦r ♥trt♦♥ ♦r ♥ ♥♠r ♦ rt♦♥sm

′

= Le t ♥t Le + 2 ♣♦ss sq♥s r ♥rt


♦rr sq♥s ♦ qm′,c t m

′

= 2 rr r♥ ♣♦②♥ s♦t♥ t c = 1 rs t tr r♥ ♣♦②♥ s ♦t♥ t c = −1

♠t♦ ♦s t♦ ♥rt st ♦ ♣♦♥ts ♦s t♦ ♥rt r♥tr②s r t ♣♦st♦♥ ♦♦r ♥trt♦♥s r st t ♣♦ss st ♦♣♦♥ts ♥t ♥ ♥♠r ♦ ♥trt♦♥s Le ♥ ♦t♥ ② ♦♥sr♥ t♦♠♥t♦♥s ♦ m

′

= 1, . . . , Le ♥ c = −1, 1 ♠ ♦ ts st♦♥ s t♦ ①♣♥ ♦ r② s t ♥ ♦♥② s♥

sq♥ s ♦♥sr ♥ s rtt♥ q r①♥ t ♦r sr♣t ♣♥♥s ♦r ts ♦ t rr

t♥♥ r②

r② ♥ ♦t♥ ② s♥ ♦♥t② t st ♦ ♦♦r ♥trt♦♥ q ♥t t st ♦ ♥trt♦♥ ♣♦♥ts t r② r s ♠♥t♦♥ ♦r t♦s sts ♥♦t ♥ ♦t♥ ♥ t s♠ ♣♥ ♥ ts ♥♥♦t s t♦tr ♥♥ts ♥♦t Π ♥t♦♥ ♣r♠tr③s t ♠♥t ♦ sq♥ ♥ rr ♦ts ♦♥s ♥ t s ♣♦ss t♦ rt t♦ ♣r♠tr③ sq♥s Π(r) ♥ Π(q) s

Π(r) = [0, α0, . . . , αLh−1, 1]

Π(q) = [0, β0, . . . , βLe−1, 1]

t αj ∈]0, ..., 1[ ♥ βk ∈]0, ..., 1[ t t♦ sq♥s r ♥ q r ♥ ♥ r♥t ♣♥s tr ①tr♠ ♣♦♥ts

rr t♦ t s♠ ♣♦♥ts p−1 ♥ pL t s r♠r tt t♦ ♦♠tr ♣r♦♣rts


♣♣r♦

s ♦r♠ t ♣r♠tr③t♦♥s ♥ r ♦♥sst♥t t♦tr s ♦♥sq♥ t s ♣♦ss t♦ tr ♥ s♦rt t♦s ♣r♠tr③t♦♥s ♥ ♦rr t♦rt ♦ ♦rr ♣r♠tr③t♦♥ sq♥ Π(r)

Π(r) = s♦rt (Π(r) ∪Π(q))

r s♦rt s ♥t♦♥ ♦rr♥ t ♠♥ts r♦♠ t ♦st t♦ t st Π(r)s ♥ ♦rr sq♥ ♦♥t♥♥ s ♦ ♦t Π(r) ♥ Π(q) ♦♥sq♥t② ♠♥t πl ♦ Π(r) ♦rrs♣♦♥s tr t♦ ♥ ♠♥t αj ♦r ♥ ♠♥t βk ♣♥♥ ♦♥t s♦rt ♦♣rt♦♥ t ts ♥ rtt♥

Π(r) = [0, ..., πl, ..., 1]

r t ♥♠r ♦ ♠♥t ♦ Π(r) s Le + Lh + 2♥ Π(r) s t ♣r♠tr③t♦♥ ♦ r② r ♥ rtt♥

r = [p−1, ...,pl, ...,pL]

t pl = [xl, yl, zl]T ♦♥sq♥t② t♦ pl ♦rrs♣♦♥s tr

♥ ♠♥t ♦ r r t zl ♦♠♣♦♥♥t s ♥♥♦♥ ♦r t♦ ♥ ♠♥t ♦ q r xl ♥ yl ♦♠♣♦♥♥ts r ♥♥♦♥ ♦♦♥ ♣r♦r ♦♥ssts ♥ tr♠♥♥ t♦s ♠ss♥ ♦♠♣♦♥♥t s ♦

pl ♠♥ts ❯s♥ t ♣r♠tr③t♦♥ Π(r) t s ♣♦ss t♦ tr♠♥ ♦r r t ♠ss♥ xl ♥ yl ♦♦r♥ts ♦ ♦♦r ♥trt♦♥s ♣♦♥ts ♦t♥ ♥st♦♥

t ♠ss♥ zl ♦♦r♥t ♦ ♥trt♦♥s ♣♦♥ts r♦♠ t r② ♦t♥ ♥ st♦♥

tr♠♥t♦♥ ♦ ♠ss♥ xl ♥ yl s

s t s ♥ s♥ t ♠ss♥ xl ♥ yl s ♦♥② ♦rrs♣♦♥ t♦ pl ♠♥ts♦t♥ r♦♠ q ♠t♦ ♦♥ssts ♥ sss② ♥ t ♠ss♥ xl ♥ yl s ♦♥ t r② strt♥ r♦♠ ♥① l = −1 ② ♦♥strt♦♥ ♠♥t πl♦rrs♣♦♥♥ t♦ ♥ ♠♥t r♦♠ Π(q) s ②s srr♦♥ ② ♠♥ts π−1 ♥πl+n ♦t ♦rrs♣♦♥♥ t♦ ♠♥ts r♦♠ Π(r) l + n s♥ts t ♥① ♦ t ♥①t♠♥t r♦♠ Π(p) ♥ t s ♣♦ss t♦ ♦t♥ xl ♥ yl t t ♣ ♦ ♥♣r♠tr③t♦♥ ♦ πl ♦♥ π−1 ♥ πl+n

xl = xl−1 +K(xl+n − xl)

yl = yl−1 +K(yl+n − yl)

tK =

πl − πl−1

πl+n − πl−1

❯s♥ ♥ t xl ♥ yl ♦♠♣♦♥♥ts ♦ pl ♠♥ts ♥ tr♠♥


♥ t ①♠♣ ♣r♦ ♥ r t ♦♦r♥ts x1 ♥ y1 ♦ t ♦♦r ♥trt♦♥ ♣♦♥t p1 s t♦ tr♠♥ ♦t ♦♦r♥ts (x0, y0) ♥ (x2, y2) ♦ ♣♦♥tsp0 ♥ p2 rs♣t② r ♥♦♥ r♦♠ t ♣r♦s r② ♦♠♣tt♦♥ ♦tt♦♠♥ r♣rs♥ts t ♣r♠tr③t♦♥ ♦ r t ♥trt♦♥ ♣♦st♦♥ pl ♦rrs♣♦♥s ♣r♠tr πl t s ts ♣♦ss t♦ ♣r♠tr③ π1 t t ♣ ♦ π0 ♥ π2 t♣r♠tr③t♦♥ rt② t ♦ K ♥ qt♦♥ ♥ ♦t ♣♦st♦♥sx1 ♥ y1 r ♦t♥ ② rs♦♥ ♥ rs♣t② ♥ tt stt♦♥ t♥♥♦♥ ♦♦r♥t ♣♦st♦♥s r rt② srr♦♥ ② ♥♦♥ ♣♦st♦♥s ♠♥sn = 1 ♦r t t♦ qt♦♥s

strt♦♥ ♦ t tr♠♥t♦♥ ♦ ♦♦r♥ts x ♥ y ♦ ♦♦r ♥trt♦♥ s♥ ♣r♠tr③t♦♥

tr♠♥t♦♥ ♦ ♠ss♥ zl s

♠r② t ♠ss♥ zl s ♦ pl ♠♥ts ♥ ♦t♥ r♦♠ r

♥ tr♠♥ s♥ t s♠ ♣r♦r s t♠ t ♦♥sr πl ♠♥ts♦rrs♣♦♥s t♦ ♥ ♠♥t r♦♠ Π(r) ♥ t s ♣♦ss t♦ ♣r♠tr③ πl ♦♥ π−1 ♥πl+n ♦t ♦rrs♣♦♥♥ t♦ ♠♥ts r♦♠ Π(q) ♥ t ♦t♥

zl = zl−1 +K(zl+n − zl)

t t ♥ ♦ ts ♦t ♦♠♣tt♦♥s t ♦♠♣♦♥♥ts ♦ ♠♥t ♦ t r② r r ♥♦♥

②s ①t♥

t♦♥ ♦ r② ♦s t♦ ♥♦ t ♣♦st♦♥ ♦ t ♥trt♦♥s ♣♦♥ts♥t♦ (O, x, y, z) ♦r ♦r t t♦♥ t t②♣ ♦ ♥trt♦♥ t t②♣ ♦


♣♣r♦

s ♥ t ♥♥t ♥ ♦t♣t ♣♥s s t♦ ♥♦♥ ♣r♣♦s ♦ tsst♦♥ s t♦ sr t r♥t ♠t♦s t♦ ♦t♥ t♦s ♥♦r♠t♦♥ ♥ tr♠♥ ♥♦tt♦♥ t♦ st♦r t♠ ♦r ♠♦r t♥ s♥ r②

♦tt♦♥ ①t♥s♦♥ Prs♦♥

♦r♥ t♦ r② s ♥ ♦rr st ♦ ♥trt♦♥ ♣♦♥ts Prt② tr②tr♥ ♠s t♦ ♥rt r ♥♠r ♦ r②s ♦♥sttt ♦ r♥t ♥trt♦♥♣♦♥ts ❯♥t tt st♦♥ ♥ ♦r t s ♦ t rr t ♥♦tt♦♥ s ♥♦t t♦♦ ♥t♦♦♥srt♦♥ tt rst② ♣r♦s ♥♦tt♦♥ ♥ ♥♦ ①t♥ t♦ ♠♥ rr②s s r② r ♥ rtt♥

rr = [pr,0, ...,pr,L]

r pr,l s t ♥trt♦♥ ♣♦st♦♥ l ♦ r② r

♥trt♦♥ ②♣ ♥

♦r ♥trt♦♥s ♣♦♥ts ♦t♥ rt② r♦♠ t s♥trs ♥trt♦♥s♣♦♥ts ♦♥ ♥ (O, x, y) ♥♦r♠t♦♥ ♦t tr ♥trt♦♥s t②♣ R|T |D ♥ trss♦t s s ♥ ♦♥sr ♦r t r♠♥♥ ♥trt♦♥s ♣♦♥ts ♦t♥ r♦♠t rt♦♥ ♦♥ t ♦♦r t s s♦ ♣♦ss t♦ ss♦t ♥ ♥trt♦♥ t②♣ R♥ s ♦r ♦♦r

♥ ♥trt♦♥ l ♦ ♥ r② r ♥ st♦r ♦t ♥t♦ s ♠tr①

Mr = [Mr,0, . . . ,Mr,l, . . . ,Mr,L−1].

♥ t②♣ ♠tr①Nr = [Nr,0, . . . , Nr,l, . . . , Nr,L−1],

Prt② ♥trt♦♥ ♦ Nr,l ♠t ♥♠r s s R = 1 T = 2 ♥D = 3 ♥ s ss♦t t♦ s ♠♥t Mr,l ♥ ♥

♥♥ ss ♦ ♥♦♠♥ ♥ t♦♥ P♥ ❲s

r② r s sss♦♥ ♦ s♠♥ts ♦♥ ② ♥trt♦♥ ♣♦♥ts pr,l ♥ pr,l+1

t l = −1, . . . , L r ♥①−1 ♥ L ♦rrs♣♦♥ t♦ t ♣♦st♦♥ ♦ t tr♥s♠ss♦♥♥t♥♥ ♥ t r♣t♦♥ ♥t♥♥ rs♣t② ♦ t lth ♥trt♦♥ s ss♦t t♥♦r♠ t♦r nr,l ♦ ②♦t s♠♥t ♥♦ ♦r s ♥♦r♠ t♦r ♥♦r♠t♦♥s st♦r ♥ Gs

sl−1 ♥ sl r t♦ ♥tr② t♦rs ♦♥ t r② ♦♥strt r♦♠ s♠♥ts [pr,l−1,pr,l]♥ [pr,l,pr,l+1] rs♣t② ♥ ♥♦♠♥ ♥ ♦t♦♠♥ ♣♥ r s♦ ♥ ♦r♥ ♥trt♦♥ l ♦ ♥ r② r s ♣t ♥ r

t ♥♥t ♣♥ s ♥ ②(

ei,⊥r,l , e

i,‖r,l

)

rr♥ ♦♥ pr,l tr♥srs t♦ sr,l−1

t ♦t♣t ♣♥ s ♥ ②(

eo,⊥r,l , e

o,‖r,l

)

♣rt♥ r♦♠ pr,l tr♥srs t♦ sr,l


♥♣t ♣♥ ♥ ♦t♣t ♣♥ ♦♥ ♥trt♦♥ l


♣♣r♦

r ei,⊥r,l ♥ ei,‖r,l r t♦ ss t♦rs ♦ t ♥♥t ♣♥ ♥ e

o,⊥r,l ♥ e

o,‖r,l

r t♦ ss t♦rs ♦ t ♦t♣t ♣♥ ♦t t♦rs ♠♥s♦♥s (3× 1)♠r② t♦ t t②♣ ♥ t ♠tr st t s ♣♦ss t♦ t ♠tr① Bi

r

tr♥ t ♥♥t ♣♥ ss ♥ ②

Bir =

[

ei,⊥r,−1, e

i,‖r,−1, . . . , e

i,⊥r,L−1, e

i,‖r,L−1

]

t ♠♥s♦♥ (3× 2L)t s s♦ ♣♦ss t♦ t ♠tr① Bo

r tr♥ t ♦t♣t ♣♥ ss ♥②

Bor =

[

eo,⊥r,0 , e

o,‖r,0 , . . . , e

o,⊥r,L , e

o,‖r,L

]

t ♠♥s♦♥ (3× 2L)♦r tr s t s s♦ ♦♥♥♥t t♦ st♦r s♣rt② t ♥♥ ♥ ♦ ♦ t

r② ♦♥ ♥trt♦♥ s ♥ s② ♦t♥ s♥ t sr ♣r♦t t♥sr,l ♥ nr,l ♥ t s ♣♦ss t♦ ♥ ♥ ♥♥ ♥ ♠tr① θr

θr = [θr,0, ..., θr,L−1]

tθr,l = arccos(nr,l, sr,l)

s s♦♥ ♥ r Bir ♠tr① tr ♥trt♦♥s r♦♠ t tr♥s♠ttr

pr,−1 t♦ t st ♥trt♦♥ pr,L−1 rs Bor ♠tr① tr ♥trt♦♥s r♦♠

t rst ♥trt♦♥ pr,0 t♦ t rr pr,L

♥trt♦♥s ♥♦ ② Bir ♥ Bo

r

♥ t♦♥ ♦♥sr♥ tt t tr♥s♠ttr ♥ rr ♥t♥♥s ♦r♥tt♦♥s ♥(O, x, y, z) ♥ ♠♦ ② t♦ tr♥s♦r♠t♦♥ ♠trs T−1 ♥ TL rs♣t②t s ♣♦ss t♦ ♥ sℓr,−1 ♥ sℓr,L t rt♦♥ ♦ ♣rtr t♦r ♥ rt♦♥ ♦rr t♦r ①♣rss ♥t♦ t ♦ ss ♦ ♦t ♥t♥♥s t

sℓr,−1 = T−1sr,−1 ♥ sℓr,L = TLsr,L

s ♦♥sq♥ t ♦ ss ss♦t t♦ t tr♥s♠ttr ♥ rr ♥t♥♥s r ♥ ②

Bℓr,−1 t ss ♦ t ♣♥ tr♥srs t♦ sℓr,−1

Bℓr,L t ss ♦ t ♣♥ tr♥srs t♦ sℓr,L

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥

s st♦♥ srs t r♥t st♣s t♦ t t tr♥s♠ss♦♥ ♥♥ r♦♠t r②s ♥ ts st♦♥ t♦r③ ♣♣r♦ s sr ♦r tt ♣r♣♦s tst♦♥ strts t sr♣t♦♥ ♦ t ♦♥♣t ♦♥sq♥t② t♦ ts sr♣t♦♥ s♣ t ♦r♥③t♦♥ r r②s sr♥ t s♠ ♥♠r ♦ ♥trt♦♥s rst♦r ♥t♦ s♠ t ♦ s ♣r♦♣♦s rtr t sr♣t♦♥ ♦ t ♥trt♦♥♦♥t t♦♥ r♦♠ t ss s sr ❲t t ♣ ♦ tt sr♣t♦♥t ♦♠♣tt♦♥ ♦r ♦s t♦ t t ♥trt♦♥s ♦ ♥ ♦ ♦ r②s st ♥ t ♥trt♦♥ ♥ t t ♣r♦♣t♦♥ ♥♥ ♥t tr♥s♠ss♦♥ ♥♥ ♥ ♦♠♣t

t ♠♥s♦♥ rr②

♦♥♣t ♦ s ♥ ♥tr♦ ♥ ❬❪ s ♥ ①t♥s♦♥ ♦ t ♠tr①♥♦tt♦♥ ♦r ♠♥s♦♥s r t♥ ♥ t ♦♣rt♦♥s tt ♥ ♦♥ s♥♠trs tr q♥t ♦r s s r r② ♦♥♥♥t strtrs st② ♥ rt② ss♠t t♦ ❵♥♠♣② ♥rr②s❵ ♦ts r♦♠ t P②t♦♥ ❵♥♠♣②❵rr② ❬❪

♦r t ♣r♦♠ t ♥ ♦♣rt♦♥s t♦ ♣r♦r♠ ♦♥ t ♠♥s♦♥rr② ♥t ♠♥s♦♥s ①s ♦s ♠♥s♦♥s r ♥♠②

rq♥② f r②s r ♥trt♦♥s l ♣♦r③t♦♥ p ♣♦r③t♦♥ q♥ rtt♥ t ♣♣rs tt rq♥② ①s ♥tr② ♣♣rs ♥ t

♥♦tt♦♥ ♦♥ t♦ t r ♥♠r ♦ rq♥s rqr ♥ ♥t ❯❲ s♥s ❯s♥ tt rq♥② ①s ♦s t♦ t q♥tt② ♦ ♥trstt♦r② t ♦♥ ♦r rq♥s ts ♦♥ t♠ ♦♥s♠♥ ♦♦♣♥

② ♦♥♥t♦♥ ①s s s♥t ② ttr ♥ rt♦♥ t ts ♥tr ♦r①♠♣ f ♦r rq♥② r ♦r r② l ♦r ♥trt♦♥ p ♥ q ♦r ♦t ♣♦r③t♦♥ss ♦♥str♥t s sr ♦r rt② ♣r♣♦s ♦r ♥ rt ♥♦r♠t♦♥ ♦t ①s s ss♦t t q♥tt② ♥♦tt♦♥s ♥ ♦♣rt♦♥ ♦♥♥t♦♥s ♦♥s s ♥ t ♦♦♥ r sr ♥ ♥♥①

♦♦s♥ ❱t♦r③ t r♥③t♦♥

♠r② t♦ ♠trs rqr♥ tt t ♥♠r ♦ ♦♠♥ ♠♥ts s t s♠♦r r♦ s rqr t♦ rs♣t ♦♥st♥t ♥♠r ♠♥ts ♦♥ ①s ♦rt ♣r♦♠ t ♥ t ♥rt r②s r♥t ♥♠r ♦ ♥trt♦♥s ♠♥s tt t② ♥♥♦t rt② st♦r ♥t♦ s ♣♦ss s♦t♦♥ t♦ ♦ tt♣r♦♠ s t♦ ss② r②s ② tr ♦♥sttt ♥♠r ♦ ♥trt♦♥s

r② s ♥ ♦rr st ♦ Ln ♥trt♦♥s s r② t t s♠ ♥♠r♦ ♥trt♦♥s Ln s st♦r ♥t♦ ♥ ♥trt♦♥ ♦ Ln ♥ r②s ♥


♣♣r♦

♦♠♣t s♥ t ♠t♦ sr ♥ t s ♣♦ss t♦ tr♠♥ sq♥(L0, ..,Ln, .,LNi−1

) ♦ Ni ♦s ♦ r②s ♦ Ln s ♦ ♦♥t♥♥ Rn r②s ♦Ln ♥trt♦♥s

r s♦s ♥ ①♠♣ ♦ tt t ♦r♥③t♦♥ ♦r r②s

Transmission Reflection Diffraction

①♠♣ ♦ t ♦r♥③t♦♥ ♦r r②s r r② s ♥trt♦♥s r st♦r ♥t♦ t Ln ♦ 0 r♥ r② s ♥trt♦♥s r st♦r♥t♦ t Ln ♦ 1 trs t ♥t♦ t Ln ♦ s♥t ♦trs r②s ♥♦t r♣rs♥t♦♥ t r

①♣rss♦♥ ♦ s ♥trt♦♥s

♦♦♥ srs t ♠t♦ t♦ t t tr♥s♠ss♦♥ ♦♥ts ♦r s♣ s

①♣rss♦♥

s ♠♥t♦♥ ♣r♦s② t ②♦t ♦♥sttt♦♥ s sr t t ♣ ♦ ♦ts ss s ss s ♥ s ♥ ♦rr st ♦ S ♦♥sttt ♠trs cs t tr ss♦t t♥ss ls

ss =

[

c0 . . . cS−1

l0 . . . lS−1

]

♠tr cc s ts s♣ ♣r♠trs s t rt ♣r♠ttt② ǫc t rt♣r♠t② µc ♦♥tt② σc r ♥ ♦ rrt♦♥ s r♦♠

nc =

√

ǫcµc.

♦r♥ t♦ ❬❪ t tr♥srs rt♦♥ ♦♥ts t t ♠tr c+1 ♦♥sr♥ ♥♥ ♥ θc ♦♥ ♠tr c s ♥ t

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥

ρ⊥c =n⊥c−1 − n⊥cn⊥c−1 + n⊥c

ρ‖c =−(n‖c−1 − n

‖c)

n‖c−1 + n

‖c

t

n⊥c =nc

cos θc

n‖c = nc cos θc

q♥t tr

♠tr ts t ♦ s♣ ♥ s ♦t♣t ♥ ♦♠s t♥ ♥♣t ♥ ♦ t ♥①t ♠tr ♥ s♦ ♦♥ t s ts ♣♦ss t♦ rt t ♥tr♠trs c‖c ♥ c⊥c ♦ ♠tr c ♦r tr♥srs ♥ ♣r ♣♦r③t♦♥ rs♣t②t

c‖c =1

1 + ρ‖c+1

[

ejδc ρ‖ce−jδc

ρ‖cejδc e−jδc

]

c⊥c =1

1 + ρ⊥c+1

[

ejδc ρ⊥c e−jδc

ρ⊥c ejδc e−jδc

]

t

δc =2πf

clcnc+1 cos θc

t♦♥

♦♥sr♥ qt♦♥ ♥ t ♥ ♦sr tt ♥tr ♠trs♣♥ ♦♥ ♦t t ♥♥ ♥ ♥ t rq♥② s ♥s t♦ t♦r sr ♥trt♦♥s t♦s ♥trt♦♥s t♦ t ♦r t s♠ rq♥②r♥ t s ♥♦t t s ♦r t ♥♥ ♥ s r♥t ♦r ♥trt♦♥ t♦ tt r♥ ♦ ♥tr t t♦ q♥tts ♥♥♦t s ♥ t s♠ ♠♥♥r♦r tt rs♦♥ ♥ t ①t♥s♦♥s ♦ c⊥c ♥ c

‖c trt t rq♥② s ♥

①s f ♥ t ♥♥ ♥ a s ♣r♠tr ♥ t♦♥ t♦ ♥ t ♣r♦ss ♦sr ♥trt♦♥s ♥ t s♠ t♠ t s rqr t♦ rt ♥ ♥trt♦♥ ①s l t♥trt♦♥ ①s ♦ rt t♦ t ♣r♠tr a t ♥trt♦♥ ♦rrs♣♦♥

s♣ ♥♥ ♥ ♥② ①t♥s♦♥ ♦ c‖c ♥ c⊥c ♥ rs♣t②rtt♥

c,ac‖f,l,p,q

c,ac⊥f,l,p,q


♣♣r♦

s rtt♥ t s ♣♦ss t♦ t t s ♥tr s ♦r ♣♦r③t♦♥

s,as‖f,l,p,q ♥ s,as

⊥f,l,p,q ♦ s s ♥ s t ♣r♦t ♦ tr ♦♥sttt ♠trs

s,as‖f,l,p,q =

0∏

u=cs−1

c=u,ac‖f,l,p,q

s,as⊥f,l,p,q =

0∏

u=cs−1

c=u,ac⊥f,l,p,q

r♥s♠ss♦♥ ♥ t♦♥ ♦♥ts

r♦♠ t s t♦♥ s t s ♣♦ss t♦ tr tr♥s♠ss♦♥ ♦rrt♦♥ ♦♥ts ♦r t ♥trt♦♥s

♦r tt ♣r♣♦s t♦ s s,aXf,l,p ♥ s,aYf,l,q s ♦♥ qt♦♥s ♥ r ♥tr♦

s,aXf,l,p =

[

1/( s,as⊥f,l,p[0],q[0])

1/( s,as‖f,l,p[0],q[0])

]

s,aYf,l,q =

[

s,as⊥f,l,p[0],q[1]

s,as‖f,l,p[0],q[1]

]

r♥ t ♥♦tt♦♥ ♦♣t ♥ ♦r (R|T |D) t ♦♥t s i=1,s,aAf,l,p,q

♥ i=2,s,aAf,l,p,q ♦rrs♣♦♥♥ t♦ rt♦♥ ♥ tr♥s♠ss♦♥ rs♣t② ♥ rtt♥

i=1,s,aAf,l,p,q = Ip,q⋆⋆ ⊙

(

sXp,1f,l sY

q,1f,l

)

i=2,s,aAf,l,p,q = Ip,q⋆⋆ ⊙ s,aX

p,1

f,l

r Ip,q s ♥ ♥tt② ♦s ③r♦s r s t♦ ♠s t ♥t♦♥ tr♠s ♦t ♣r♦t ♥ ♥ ♥ ②

Ip,q =

[

1 00 1

]

t♦♥ ♦ ♥trt♦♥s

s ♣rt srs t r♥t st♣s t♦ t t ♥trt♦♥s ♦ ♥♥trt♦♥ ♦ Ln t t ♣ ♦ sr♣t♦♥ ♦r tt ♣r♣♦s rst♣rt ts t ♣r♦r t♦ r♦♠ t ♠t r② ♥♦r♠t♦♥ ♦t♥ ♥ ♥ ♣rtr t ①t♥s♦♥ t♦ ♦ t ss ♥ ♥trt♦♥s ♠trs s①♣♥ ♥ t r♥t rs♣♥ ♦♣rt♦♥s ♦♥ t t♦r③ t♦♥ ♦t ♥trt♦♥ s t ♥ tt t♦♥ ♣r♦r♠ t ♥rt rs♣♥♦♣rt♦♥s r ♣♣ ♦r t ♣♦♠♥ ♥♥ st♠t♦♥

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥

t ♠♥s♦♥ rr② st♦r

♣♦♠♥ t♦r③ ♦♠♣tt♦♥ ♦ t ♥trt♦♥s rst rqrs tt tss s ♥ ♥trt♦♥s ♠trs ♦♥rt ♥t♦

ss

①♣rss♦♥ ♦ ss ♠trs Bir ♥ Bo

r s ♥ ♦t♥ ♦r s♥ r② ♥ ♦s ♠trs ♠♥s♦♥ (3×2L) r L s t ♥♠r ♦ ♥trt♦♥s♦ t r② ♦r♥ t♦ t t ♦r♥③t♦♥ sr ♥ sst♦♥ r②t Ln ♥trt♦♥s ♦♥s t♦ Ln ♦ ♥ ♥ ♦rr t♦ tr♥ t s ♠trs ♥t♦s t s♠ ♦r♥③t♦♥ s t♦ ♦s♥ ♦t s s ♥ ts rtt♥♦r Ln ♦

inBr,lp,x

onBr,lp,x

r t ①s r x ♥ lp r♣rs♥ts t r② ①s s♣ ①s ♥ lp t ♦♥t♥t♦♥♦ ♦t ♥trt♦♥ ♥ ♣♦r③t♦♥ rs♣t② t ♠(r) = Rn ♠(x) = 3 ♥♠(lp) = 2Ln ♥ t ①s lp ♥♦t t t tt t ss t♦rs ♦ ♥trt♦♥s ♥ st ♥t♦ t s♠ ①s t s ♣♦ss t♦ r♥tt t♦st♦ ①s s♥ t rs♣ ♦♣rt♦♥ s ♥ ♥

inBr,l,p,x = rs♣

(r,l,p,x)

(

inBr,lp,x

)

onBr,l,p,x = rs♣

(r,l,p,x)( onBr,lp,x)

t♦ ①s l ♥ p r♣rs♥t t ♥trt♦♥ ♥ ♣♦r③t♦♥ ①s rs♣t② t♠(l) = Ln ♥ ♠(p) = 2

ss♠♥ t ♦♥♥t♦♥ tt t rst ♥ st ♥trt♦♥ ♦ r②s r ss♠t t♦ t tr♥s♠ttr ♥ t rr rs♣t② t s ts ♣♦ss t♦ ♥ ♥♥tr② nBr,l,p,q ♥♥ t♦ sss ss ♠trs t

nBr,l,p,q =onB

x,pr,l

inB

x,pr,l

s t ♦ ss ♠trs ♦ t ♥t♥♥s Bℓ−1 ♥ Bℓ

Ln♥ rrtt♥

♥t♦ t♦ s ℓBl[−1],x,p ♥ ℓBl[Ln],x,p rs♣t② t ♦s t s♠ ♦r T−1 ♥TLn ♦♠♥ Tl[−1],x ♥ Tl[Ln],x rs♣t②

rt♥ ♥ ♥trt♦♥s s

r♦♠ ♥ r② r♥t t②♣s ♦ ♠t ♥♦r♠t♦♥ ♥ ♦t♥ ♥t s ♥ s♦♥ tt t♦s ♠t ♥♦r♠t♦♥ r st♦r ♥t♦ ♥ ♥trt♦♥ t②♣♠tr① Nr s ♠tr① Mr ♥ ♥ ♥♥ ♥ ♠tr① θr ♦s ♠trs rtr♥rs♣t② t ♦ ♥trt♦♥ t②♣ t ♥trt♦♥s ♠tr ♥ t ♥♥♥ ♦r ♥trt♦♥s ♦ ♥ r② ♠r② t♦ ss ♠trs t s ♣♦ss t♦rt ss♦t s ♦r ♥ Ln ♦ nNr,l nMr,l ♥ nθr,l


♣♣r♦

② ♦♥strt♦♥ nNr,l nMr,l ♥ nθr,l r s rrr♥ t♦ t s♠ ♦ ts♠ r②s ♥ t s♠ ♥trt♦♥s ♥♦r♠t♦♥ ♦ t♦s s ♥ ts ♥trt ♥t♦ s♥ ♥trt♦♥ ♥ ②

n,i,s,aAr,l

r i s ♥ a ♣r♠trs ♥t t t②♣ ♦ ♥trt♦♥ t s s ♥ t♥♥t ♥ ♦♥ ♥trt♦♥ l ♦ r② r ♥ ♦ n rs♣t② ♣r♠trs i s ♥a r rs♣t② rr ② nNr,l nMr,l ♥ nθr,l

s♣♥ ♥ ss②♥ s

♥ ♦rr t♦ ①♣♦t t t♦r③ ♦r♠t ♦ t s ♥trt♦♥ ♦ s♠t②♣ ♥ ♥♦♥ t s♠ s r t ♥t② s ♦ s t♦♥rqrs t♦ s♣t t ♦ ♥t♦ sr s♠r s Prt② t ♦♥ssts♥ st♥ s ♠ s s t② r ss ♥ ♥trt♦♥ t②♣s ♦ ttsst♦♥ st♣s r rqr rst t ♦ n,i,s,aAr,l r r♦r♥③ s♦♥ tsst♦♥ s ♣♣ ♥ tr♠s ♦ ♦♠♣tt♦♥ s♣ ♦t st♣s r q t② ♦♥②r② ♦♥ ♠♠♦r② rrss♥ ♣r♦ss

t r♦r♥③t♦♥ ♦ n,i,s,aAr,l ♦♥ssts ♥ rs♣♥ t ② ♦♥♥ tr② ①s ♥ t ♥trt♦♥ ①s

n,i,s,aArl = rs♣(rl)

( n,i,s,aAr,l)

r rl s dim(rl) = RnLn♥ t♦ ♦♥t♥t t ♦s n ♦♥ t rl ①s

i,s,aArl =|

rl|

N−1

n=0n,i,s,aArl

r rl s dim(rl) =∑N−1

n=0 RnLnss♠♥ t ♦♥♥t♦♥ tt rss♥ ♥ ♥trt♦♥ ♦rrs♣♦♥♥ t♦ s♣

s ♦r s♣ ♥trt♦♥ t②♣ rqrs s♣②♥ ♣r♠tr s ♦r i rs♣t②♦♥sr♥ tt t ♥trt♦♥ ♥♦s r♥t t②♣ ♦ ♥trt♦♥s ♥ St②♣ ♦ r♥t ss t s ♣♦ss t♦ r i,s,aArl r♦♠ qt♦♥ s

i,s,aArl =3⋃

u=1

sS−1⋃

v=s0

i=u,s=v,aArl

♦s rs♣♥ ♥ sst♦♥ ♦♣rt♦♥s ♦♥ t ♥trt♦♥ r r♣rs♥t ♦♥ r

t♦♥ ♦ ♥trt♦♥s

r♦♠ tt ♣♦♥t t ①sts sr s♠ s ♦r♥③ ② s s ♥ ♥trt♦♥t②♣ i s ♦r♥③ ♦♥ ♦ t♦s ♥ t t t ♣ ♦

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥


♣rt♦♥s ♦♥ t ♥trt♦♥ trt♥ r♦♠ t ♦r♥③ ♥ Ln♦ t ♦♥t♥t♦♥ ♥ rs♣ ♦♣rt♦♥ ♥ t s♣t♥ ♥t♦ sr ♣♥♥ ♦♥ t ♥trt♦♥ t②♣ ♥ ♥♦ s

t qt♦♥s ♥ t♦♥ ♣r♦ss ♠s t♦ rtr♥ rt♦♥tr♥s♠ss♦♥ ♦r rt♦♥ ♦♥t ♦r ♥ rq♥② r♥ f rst♥s s ts ♥r t ♦r♠

i,sAf,nrl,p,q

s♣♥ ♥trt♦♥s

♥ t s t♦♥ s ♥ ♣r♦r♠ t ♦r♥ ♦ ♦r♥③t♦♥ ♦ tr②s s t♦ rtr ♦r t ♣♦♠♥ ♥♥ t♦♥ s t ♣♣rs ♦♥ r t♦s rs♣♥ ♦♣rt♦♥s r ♦s t♦ t♦s sr ♥

t ts ♣♦♥t tr s s ♠ t ♥trt♦♥ s s s ♥ t②♣ ♦♥trt♦♥ rst st♣ s t♦ tr t♦s ♥trt♦♥ s ♥t♦ s♥ ♦♥t t ♣ ♦ t ♥♦♥ ♦♣rt♦r

Af,rl,p,q =⋃

i,s

i,sAf,rl,p,q

♦ t ♦♠♣t t ♥trt♦♥ ♥ rs♣ t♦ ts ♦r♥ s♣s ♥ ♠ ♥ t♦ st♣s rst ② rrt♥ ♦s ♦ ♥trt♦♥ t

nAf,rl,p,q =−1

|

rl|

N

n=0

(Af,rl,p,q).


♣♣r♦


♣rt♦♥s ♦♥ t t ♥trt♦♥ t s rs♣ t♦ ♦t♥ t ♦r♥ ♦r♥③t♦♥ ♥ Lnblock

♦♥ ② rs♣♥ ♦ t t ♦r♥ r② ♥ ♥trt♦♥ ①s

nAf,r,l,p,q = reshape(f,r,l,p,q)

( nAf,rl,p,q).

Pr♦♣t♦♥ ♥♥ ♥ r♥s♠ss♦♥ ♥♥

♥♦ ♦ t t ♥trt♦♥s ♦s t♦ rt t ①♣rss♦♥ ♦t ♣r♦♣t♦♥ ♥♥ r♦♠ tt ♣♦♥t t sr r② tr♥sr ♥t♦♥ ♦ t tr♥s♠ss♦♥ ♥♥ ♥ r ② ♥ t ♥t♥♥ sr♣t♦♥s ♥ t ♦♠♣tt♦r③ ①♣rss♦♥ ♦ t tr♥s♠ss♦♥ ♥♥ s ♦t♥

Pr♦♣t♦♥ ♥♥

♣r♦♣♦s ①♣rss♦♥ ♦ t ♣r♦♣t♦♥ ♥♥ ♥♥ t♦t ♥t♥♥s s♣t r♦♠ ❬❪ t♦ t ♥t♦ ♦♥t t ♥♦tt♦♥ ♥ ts ♣r♦r♠ ♦♠♣tt♦♥ ♦r r r♥ ♦ rq♥s s♠t♥♦s② t ♥♦tt♦♥ s s t♦t tt t ①♣rss♦♥ ♦♥② ss♠s t tt♥t♦♥ ♥ ♣♦r③t♦♥ ②s r♥tr♦ r♥ t ♦♠♣tt♦♥ ♦ t tr♥s♠ss♦♥ ♥♥ ♥♥ ♥♥ ♥t♥♥s ❯s♥ ♥ t ♣r♦♣t♦♥ ♥♥ abn Cf,r,p,q t♥

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥

♣♦♥ts a ♥ b ♥ rtt♥ s t ♦♦♥ ♣r♦t

abn Cf,r,p,q = nB

p,q

⋆,r,l[Ln]

0∏

u=ln−1

nAp,q

f,r,l[u] nBp,q

⋆,r,l[u]

♥ t ♣r♦t ♦♥ t ♥trt♦♥s s ♥ ♣r♦ss t abn Cf,r,p,q

♦s ♥♦t ♣♥ ♦♥ t ♥♠r ♦ ♥trt♦♥s s ♦♥sq♥ abn Cf,r,p,q r♦♠ ♦ n ♥ st ♦♥ s♥ abCf,r,p,q ♦♥ t r ①s

abCf,r,p,q =

[

ab0 Cf,r,p,q, ..,

|r|, ..., abN−1Cf,r,p,q

]

abCf,r,p,q =|r|

N−1

n=0

abn Cf,r,p,q

r t ♠♥s♦♥ ♦ ①s r s ♥♦ dim(r) =∑N

n=0 n × Rn t♦s st♣s rstrt ♥ r

r♠ s♦♥ ♦ t♦ ♦t♥ t ♣r♦♣t♦♥ ♥♥ abℓCf,r,p,q ♦r t rq♥② r♥ r♦♠ t ♥trt♦♥s s ss ② Ln ♦s


♣♣r♦

♥② t ♥ ♦sr tt ts ①♣rss♦♥ ♦ t ♣r♦♣t♦♥ ♥♥ ♦s ♥♦t♣♥ ♦♥ t ♥t♥♥ ♦r♥tt♦♥s ② ♥♥ t ♦ ss tr♥s♦r♠t♦♥ aℓBp,q

♥ bℓBp,q ♦ t tr♥s♠ttr ♥ rr rs♣t② t

aℓBp,q =ℓB

x,pl[−1]

(

Tx,p

l[−1]Bp,q

l[−1]

)x,q

l[−1],

bℓBp,q =ℓB

x,pl[Ln]

(

Tx,p

l[Ln]Bp,q

l[Ln]

)x,q

l[Ln],

t s ♣♦ss t♦ ①♣rss t ♣r♦♣t♦♥ ♥♥ ♥ t ♥t♥♥s ♦ ss abℓCf,r,p,q

tabℓCf,r,p,q =

bℓ Bp,q⋆,⋆

(

abCp,q

f,raℓB

p,q⋆,⋆

)p,q

f,r

r ② r♥sr ♥t♦♥

❲t t ♣ ♦ t ♣r♦♣t♦♥ ♥♥ ①♣rss♦♥ t s ♣♦ss t♦ rtt t♦ s ♦ r ♥♦♠♥ bℓEf,r,p t♦ t tr♥s♠tt ♦t♦♥ aℓEf,r,p r ℓ ♥♦ts tt t s ①♣rss ♥ t ♥t♥♥s ♦ ♦♦r♥ts②st♠s

bℓEf,r,p =abℓC

p,q

f,raℓE

p,1

f,r

♥ t♦ tr♠♥ t tr♥s♠tt ♦t♦♥ t s ♣♦ss t♦ ♥ t♦ q♥tts ♦r t rr ♥ t tr♥s♠ttr

aFf,r,p =√

aGf,⋆,paUf,r,p

bFf,r,p =√

bGf,⋆,pbUf,r,p

r aGf,p ♥ bGf,p r t ♥ ♦ t ♥t♥♥ a ♥ b rs♣t② ♥ ♥ rt♦♥asr bsr s aUf,r,p ♥ bUf,r,p t ♣♦r③t♦♥ stt ♦ t ♠ttr♥ tr♥s♠ttr ♥t♥♥ rs♣t② ♦t tt ♦r ♦♠♣tt♦♥ ♥② ♣r♣♦s♥t♥♥s ♦ t ♥t ♦ t s♣r r♠♦♥ sr♣t♦♥ s ♣r♦♣♦s ♥❬❪ aUf,r,p s ①♣rss ♥ t ♦ ss ℓBr,l[Ln],x,p ♦r ℓBr,l[−1],x,p ♣♥♥ ♦♥ t rrs t♦ t ss♦t t t ♥t♥♥ ♦ tr♥s♠ttr ♦ rr rs♣t② ♠tt tr aℓEf,r,p ♥ ♦t♥ t

aℓEf,r,p =

√

2Z0

4π

(

1− aS1,1f

)

aF1,pf

r Z0 s t r s♣ ♠♣♥ ♥ aSf s t sttr♥ ♣r♠tr rtr③♥t ♠s♠t♥ ♦ t ♥t♥♥ a t s ts ♣♦ss t♦ rt t r② tr♥sr ♥t♦♥s

abHf,r =−jc√

4π√2Z0ff,⋆

(

1− bSf,⋆)

bFp,1

f,rbℓE

p,1

f,r

♦♠♣tt♦♥ ♦ t r♥s♠ss♦♥ ♥♥

r bSf s t sttr♥ ♣r♠tr ♦ t ♥t♥♥ b ♥② ② ♥♥

abγf =−jc4πff

(

1− bS1,1f

)(

1− aS1,1f

)

abαf,r =bF

p,1

f,r (abℓC

p,q

f,raF

p,1

f,r)p,1

f,r

t♥ qt♦♥s ♥ ②

abHf,r =abγf,⋆

abαf,r

t ♠② ♥♦t tt abγf ♦♥② ♣♥s ♦♥ t sttr♥ ♣r♠trs ♦ ♥t♥♥s t ♦t①tr♠ts

♦♠♣t r♥s♠ss♦♥ ♥♥

♦r ♥♥ tr♥sr ♥t♦♥ s r r♦♠ t s♠♠t♦♥ ♦ ♦♥trt♦♥s♦♠♥ r♦♠ r② ♥ ② r♥tr♦♥ t ♣r♦♣r ♣s tr♠

abHf =abHrfexp(−2jπff,⋆τ

r⋆ )

abHf =abγfabα

rfexp(−2jπffτ

r⋆ )

t r♦s ♦ ♥t♥♥s r st t tr♥sr ♥t♦♥ ♦ t tr♥s♠ss♦♥ ♥♥♦♠s

baHf = baγfbaα

rfexp(−2jπffτ

r⋆ )

t

baγf = abγf baαf,r =

abαf,r

❲ ♥

baαf,r =aF

p,1

f,r (baℓC

p,q

f,r,bF

p

f,r)p,1

f,r

= bFp,1

f,r (abℓC

p,q

f,r,aF

p

f,r)q,1

f,r

♥♦♥ tt t Af,r,l,p,q s s②♠♠tr Af,r,l,p,q = Af,r,l,q,p t tr♥s♣♦s ♦abCf,r,p,q ♥ ② t qt②

baCf,r,p,q =ab Cf,r,q,p

♥ baℓCf,r,p,q =abℓ Cf,r,q,p

♣r♦s t s♦♥ qt② ♥ s t ♥♥ r♣r♦t② s r

baHf,r,p,q =ab Hf,r,p,q


♣♣r♦

t ♣ t ♦

s s♥ ♥ ♣r♦s st♦♥ t s ♦ t ♦s t♦ ♦♠♣t t ♦♠♣t♥♥ ♠♣s rs♣♦♥s ♦ ♥ ②♦t r♦♠ tt ♦♠♣tt♦♥ t s ts ♣♦sst♦ ①trt ♦srs ♥ ♦♥strt Ps rqr ♦r ♥ ♦③t♦♥ ♦rt♠ss♥ t ♠t♦ sr ♥ ❬❪

♥ ♦ t ♠♥ ♣♦♥t ♥♥ t t♦r③ ♣♣r♦ ♦ t s t sr♣t♦♥♦ t ss s♥ s ♥ s sr♣t♦♥ ♦s t t♦♥ ♦ ♥trt♦♥♦♥ts ♦r sr ♥trt♦♥s ♥ r r♥ ♦ rq♥s s♠t♥♦s② ♥ tt s♣ ♠t sr♣t♦♥ s t♦ t ♥t ♦ t t♦r③sr♣t♦♥ ♥♦t ♦r t♥ t ♦♥ts ♦♥ r ♥♠r ♦ r②s ♦t♥ r♦♠ tr♣s t ♦♥② ♦♥ t rt ♣t t♥ t tr♥s♠ttr ♥ rr ♥ ♦♥sr♥♦♥② t tr♥s♠ss♦♥ ♦♥ts

rst s♦rt ♦r ♦ t ♠♦st ♦♠♠♦♥ t ♠♦s ♥ ♦rr t♦ tt r♥s t t ♣r♦♣♦s ♠♦ s ♣rs♥t rtr ♦t t ♠t♦ t♦tr♠♥ t ♣♦r ♦sss ♥ s♣ ♣r♦ss t♦ t t ② t♥ ttr♥s♠ttr ♥ t rr r sr ♥② t ♦♠♣tt♦♥ ♦ t ♥trt♦♥♦♥ts s strt t ss ♥ ♠trs sr ♥ ♣r①st♥ ♦r

①st♥ t ♠♦s

t ♣♣r♦s r sttst ♠t♦s ♦♥ssts ♥ t♥ t ♥r②♦sss ♦♥ t rt ♣t t♥ t♦ ♥ts t♦♥ ♦ t ♦sss ♥ ♠t r♥t ♠♦s

♦t② ♥♥ ♦ ♦ t ❲ ♥ ♦♦r ❲❲t t ♣ ♦ ♣t ♦ss s♦♥ ♠♦ t ♠t ♣♣r♦s ♠s t♦

♣rt t ♦sss ② ♥ s♦♠ rs♠ t t ♥tr♦t♦♥ ♦ ♦♦r ♣♥ ♥♦r♠t♦♥ ♦t② ♥♥ ♠♦ ❬❪ t ♦sss r st♠t ♦♥ rt ♣t t♥ tr♥s♠ttr ♥ rr t♦t ♦sss r st♠t ② ♥ t♦ t r s♣♦sss ♦ss tr♠ t♦ t ♦sts s ♦♦rs ♠♦s ❬❪ s ♥ ♠♣r ♦rrt♦♥ tr♠ t♦ t ♦t② ♥♥ ♠♦ t♦ t ♥t♦ ♦♥srt♦♥t t②♣ ♥ s ♠t ♦♦r ♦♥srt♦♥s ♠♦s ❬❪ ♥tr♦ ♥♦♥ ♥r ♦rrt♦♥ t♦r t♦ t ♠♦ s ♥t♦♥ ♦ t ♥♠r ♦r♦ss s ♥ ♥② ❲ ♠♦ ❬❪ s t♦ s ♠♦s t r♥tt♦♥♦ ♦t t ♦sss ♥tr♦ ② t t♦ t ♦sss ♥tr♦ ② t ♦r t♦♦r ♦♠♠♦♥ t♥ t♦s ♠♦s s t♦ rqr ♣rs sr♣t♦♥ ♦t ♥r♦♥♠♥t t♦ ①t② ♥♦ t t②♣ ♦ ♠tr r♦ss ② t rt ♣t ♥①♣♦t♥ st♦st ♥♦r♠t♦♥ ♥ ♦rr t♦ t t ♦sss

♦sss st♠t♦♥s

❯♥ t ①st♥ ♠♦s t ♣r♦♣♦s ♠t ♣♣r♦ ①♣♦ts t ♣rs♥♦ ♦ ♦t t ♥r♦♥♠♥t r♦♠ t r♣ sr♣t♦♥ ♥ st♦♥ ♥

t ♣ t ♦

t tr♠♥st s sr♣t♦♥ ♥ st♦♥ t♦ st♠t t ♦sss r♥tst♣s sr♥ t ♦sss t♦♥ s ♥ ② ♦rt♠ s rst st♣ t r♣sr s t♦ tr♠♥ ♥ ♦rr st ♦ s♠♥ts r♦ss ② t rt ♣t t♥t rr ♥ t tr♥s♠ttr r♦♠ tt st ♦ ♥trst s♠♥ts t s ♣♦ss t♦♦t♥ ♦t t ♦rrs♣♦♥♥ ♦rr st ♦ ♥♥ ♥s θ = [θ0, . . . , θLi−1] ♥t ♦rr st ♦ ♥trst ss M = [M0, . . . ,MLi−1] r Li s t ♥♠r ♦♥trst s♠♥ts ♥ s♥ t tr♥s♠ss♦♥ qt♦♥ t ♦tt st ♦ ♥♥ ♥s θ ♥ t st ♦ ♥trst ss M t s ♣♦ss t♦ tt q♥t tr♥s♠ss♦♥ eAf,p,q ♦r t Li ♥trst s♠♥ts t

eAf,p,q =

0∏

l=Li−1

i=1,s=Ml,a=θlAf,l,p,q

♦sss ♦♥ t rt ♣t Pf,p,q ♥ ts ♦t♥ ♦r ♦t ♣♦r③t♦♥♥ ♦r sr rq♥s t

Pf,p,q = −20 log10 | eAf,p,q |

♦rt♠ ♦♠♣t ♦sss s♥ t t❲ ♠♦

qr t, r,Gs⊲ t st ♦ s♠♥ts ♥trst ② t rt ♣t t♥ t, r ♥ t ss♦tst ♦ ♥♥ ♥ la ♥ lsθ,S = s♦♥♥(t, r,Gs)⊲ ♥ t st ♦ ♠trs M ♦rrs♣♦♥♥ t♦ t s♠♥ts SM = s♠t(S)⊲ t t tr♥s♠ss♦♥ ♣r♠trs ♣r ♥ ♦rt♦♦♥ ♦r t r♦ss♠trs ♦r ♥ rq♥② r♥ f

Po,Pp = (θ,M, f)⊲ ♦sss r ♦t♥ ♦r ♦t ♣♦r③t♦♥Lo = −20 log10 sum(Po)Lp = −20 log10 sum(Pp)

①ss ② st♠t♦♥

② ♦♥strt♦♥ t q♥t tr♥s♠ss♦♥ ♥ ♥ s t♦♠♣① s ♥ t tr♥s♠ss♦♥ ♥tr♥s② ♦♥t♥s ♥♦r♠t♦♥ ♦tt ①ss ② τe ♥tr♦ ② r♦ss♥ t ss r♦♠ ts ♥♦r♠t♦♥ t s♣♦ss t♦ ♦♠♣t t ♦ t ♦ ② τ ♦sr ② t rr ts ♥t♦ ♦♥t tt ①ss ② ♦ ② τ ♥ rt② ♦t♥ ②s♠♠♥ t ①ss ② τe ♥ t ② ♦rrs♣♦♥♥ t♦ t t♠ s♣♥ ♥ r s♣♣r♦♣t♦♥

Prt② t ② ss♦t t♦ t r s♣ ♣r♦♣t♦♥ s ♥♦t s② sss♦ t ② ss♦t t♦ t st♥ t♥ t tr♥s♠ttr ♥ t rr τLOSs ♣rr② s ♦r t s ♦ τLOS rqrs s♦♠ ♦♥srt♦♥s


♣♣r♦

rt r② trt♦r② tr♦ t ♠tr ♥t♥ t rrt♦♥ ♥st ♠tr

♥ ♠♥♠ ①♠♣ ♦ r t ♥ ♦sr tt r♥ ①sts t♥ t ② ♦ t rt ♣t τLOS ♥ t ② ♦ t r s♣ s r♥ st ② τM ♥tr♦ ② t t♠ s♣♥ t♦ r♦ss♥ t s s ② s ss♦tt♦ t st♥ dl ♥ ♥ ♦♠♣t t

dl = dMlcos(θl)

r Ml ♥ θl r t s t♥ss ♥ t ♥♥ ♥ ♦ t lt ♥trt♦♥♥ t ♥trt♦♥ s r♦ss ♦rrs♣♦♥s ♥ ①ss st♥

♥ t s ♣♦ss t♦ ♥ dl ♦♥t♥♥ t ①ss st♥s ♥tr♦② t ss ♦♥ t rt ♣t ❯s♥ tt t ② τM ♥ rt②♦♠♣♥st r♥ t q♥t tr♥s♠ss♦♥ ♦♠♣tt♦♥ ♦♠♣♥stq♥t tr♥s♠ss♦♥ cAf,p,q ♥ ts rtt♥

cAf,p,q =

0∏

l=Li−1

i=1,s=Ml,a=θlAl,1f,p,q exp(jd

l,1⋆,⋆,⋆)

ss♦t ♦♠♣♥st ①ss ② τc(f , p) ♦r ♥ rq♥② f ♥ ♥♣♦r③t♦♥ p ♥ ♦t♥ r♦♠ cAf,p,q t

τc(f , p) =

arg( cAf [f ],p[0],q[0] ), ♦r ♥ ♦rt♦♦♥ ♣♦r③t♦♥

arg( cAf [f ],p[1],q[1] ), ♦r ♥ ♣r ♣♦r③t♦♥

♥② t ♦ ② τ(F, P ) ♣♥s ♦♥ t rq♥② ♥ ♣♦r③t♦♥ ♥ s♦t♥ t

τ(f , p) = τLOS + τc(f , p)

t ♠② ♥♦t tt t ♠♦ ss♠s ♣r♦♣t♦♥ ♦ t r② ♦♥ strt♥ ♥s t s ♥t♥ t rrt♦♥ ts s♥ r♥t r② ♦t♣t ♣♦st♦♥ s ♦♥sq♥ ♥ rr♦r ♥ t ♦♠♣t Ps s s ♥tr♦ s s

t ♣ t ♦

s t ♠tr s t ♦r s♦r♥t ♠t♦ s♦ srs t s♠ ♠tt♦♥st♥ ♥② ♠t ♠♦ s t ♣♦r ♥ ② ♥♦r♠t♦♥ r ♦♥② ♦t♥r♦♠ t rt ♣t t ♦♠♣t Ps ♥♥♦t t t r ♦♥s ♥ s ♦ ♣ stt♦♥ r t s ♥♦t ♦sr ♦r t s♠t rsts ♦♠♣rt♦ ♠sr♠♥ts ♥ st♦♥ ♣r♦ t ♥trst ♦ t ♠t♦

t♦♥ ♦ t t♦♥ ♥ r♥s♠ss♦♥ ♦♥ts

t♦♥ ♦ t tr♥s♠ss♦♥ ♥ rt♦♥ ♦♥ts s ♠♥ ♣rt ♦ ♦tt ♥ t st s♣ ♠t ♣♣r♦s t ♥srs t ♦rrt st♠t♦♥♦ t Ps ♦r tt ♣r♣♦s ts ♣rt ♣r♦♣♦ss t♦ strt ♥ t t st♦♥ t ♥ ①♠♣ ♦rr♦ r♦♠ ♣r ①st♥ ♦r ❬❪ s ①♠♣♦♥srs s ♦♠♣♦s ♦ s♥ ♦♠♦♥♦s ♠tr ♦r r♥t t♥sss5cm 10cm ♥ 15cm s ♠tr s ♣r♠tt② ǫr = 3 ♥ ♦♥tt②σ = 0.06S/m rq♥② s ♦s♥ t 4 GHz

♦t♦♥ ♦ t rt♦♥ ♦♥t ♥ ♣r ♥ ♦rt♦♦♥ ♣♦r③t♦♥ s ♥t♦♥ ♦ t ♥♥ ♥ s s♦♥ ♥ r ♥ rs♣t② s♠ r♣rs♥tt♦♥ s ♦♣t ♦r t tr♥s♠ss♦♥ ♦♥t ♣rs♥t ♥ r ♥ ♦s rsts r ♦♥sst♥t t t rsts ♣rs♥t ♥ ❬❪

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rt♦♦♥ ♣♦r③t♦♥

♦ ♦ rt♦♥ ♦♥ts s ♥t♦♥ ♦ t ♥♥ ♥ ♦r ♣r ♣♦r③t♦♥ ♥ ♥ ♦rt♦♦♥ ♣♦r③t♦♥ t ǫr = 3σ = 0.06S/m ♥f = 4 GHz

s ♥ ①♠♣ t ♠♦s ♥ ♣s ♦ t rt♦♥ ♦♥t ♥ stt♥ 2 GHz ♥ 16 GHz ♦♥sr♥ ♠tr t ♦♠♣① ♣r♠tt② ♥ rtt♥ t ǫ = ǫ′+ jǫ′′ t♦♥ s ♥ ♣r♦r♠ t ♠tr♦ 5 cm ♦r ǫ′ = 2, 3, 4 ♥ ǫ′′ = 0.04, 0.1 rs ♥ s♦ t ♠♦s♥ ♣s rs♣t② ♦r r ♣rt ǫ′ = 0.03 ♦r tt t ♣r♦♣t♦♥ ② s0.57 ns s ♦♠♣♥t t♦ t ♦st♦♥s ♦ GHz ♦sr ♥ rs ♥ s♦ t ♠♦s ♥ ♣s rs♣t② ♦r r ♣rt ǫ′′ = 2


♣♣r♦

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rt♦♦♥ ♣♦r③t♦♥

♦ ♦ tr♥s♠ss♦♥ ♦♥ts s ♥t♦♥ ♦ t ♥♥ ♥ ♦r ♣r ♣♦r③t♦♥ ♥ ♥ ♦rt♦♦♥ ♣♦r③t♦♥ t ǫr = 3σ = 0.06S/m♥ f = 4 GHz

♦♥s♦♥

tr r stt ♦ t rt ♦♥ t ①st♥ t♦♦s ♥ s♦♠ ♠t♠t ♠♥ts ♦♥ t ♣r♦♣t♦♥ ♥♥ ts ♣tr s ♣rs♥t r♣ s t♥q rst ♣rt s ♥ ♦s ♦♥ t sr♣t♦♥ ♦ t ♥r♦♥♠♥t s♥ r♣s♥ t s ♥ s♦♥ ♦ tt r♣ s sr♣t♦♥ ♦ s♦ ♥t♦s② s ♦r sr♥ t tr♦♠♥t ♥trt♦♥s ♥ ♣rtr t s ♥ s♦♥♦ ts sr♣t♦♥ ♦ ♥t♦s② s ♦r ♦♣♥ ♥ ♥r♠♥t ② ♥tr♦♥ t ♥ ♦♥♣t ♦ r② s♥tr s♦♥ ♣rt s s♦♥ ♦ r②s♥ ①trt r♦♠ tt r♣ sr♣t♦♥ s♥ ♠s ♥ ♣r♦t♦♥s ♠t♦s ♥r♥t ♣♥s ♥ ♦ strtr ♠♥t ♥ ①trt r♦♠ t r♣ ♦r t♣♦♠♥ t♦♥ ♥ tr ♣rt ♥ ♦r♥ t♦r③ ♦r♥③t♦♥ ♦ t t♦♠♥ t♦ ♥ ♦r♠s♠ s ♦♥ s ♥ ♥tr♦ s ♦r♠s♠♦s ♥ rst t♠ st t♦♥ ♦ t ♥trt♦♥s r♦♠ t s♠tr②s ♥ t ♦♠♣t st♠t♦♥ ♦ t ♣r♦♣t♦♥ ♥♥ ♦r r ♥♠r ♦rq♥② ♥ t s♠ t♠ ❯s♥ t s♠ ♦r♠s♠ t ♣r♦♣t♦♥ ♥♥ s s♦t ♥ st s♣ ♠t ♣♣r♦ s ♥ ♣rs♥t s ♣♣r♦ts ♥t ♦ t t♦♥ ♦ t ss t♦ q② ♣r♦ Ps ♦ ♥rt ♣t ② t♥ ♥t♦ ♦♥t t rst sr♣t♦♥ ♦ ss ♥② t s♦♠♣tt♦♥ s ♥ ♦♠♣r t♦ ♥ ①st♥ ♠♦

♦♥s♦♥

2 4 6 8 10 12 14 16Frequency (GHz)

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t♦♥ ♦♥ts ♥ ♣r ♣♦r③t♦♥ ♦r ♦♠♣① ♣r♠tt② ♦♥t ♥♥ ♦ t r ♣rt t ① ♠♥r② ♣rt ǫ′′ = 0.03 s s♦♥ ♥ ♠♦s ♥ ♣s ♥♥ ♦ t ♠♥r② ♣rt t ① r ♣rtǫ′ = 2 s s♦♥ ♥ ♠♦s ♥ ♣s

♣tr

♥♦♦r ♦③t♦♥ ♦st ♥tr♦♥♦s ♣♣r♦s

♥tr♦t♦♥

r♥ t st ②rs t ♦③t♦♥ ♥♦r♠t♦♥ s ♠♥② ♥t t♦ ♥ srs♦ rss ♥t♦rs ♦r ♦t t ♥rs ♦ t rt rqsts ♥ t ♥ t♦r ♥r② ♦♥s♠♣t♦♥ ♦♣ ♥ ♣r♦♠ts ♥ rss ♥t♦rs ttt ♣♦st♦♥ ♥♦r♠t♦♥ ♦ ♣ t♦ rs♦ ♥ ♠♦st ♦ t ♥s s♠s♦ rss ♥t♦rs ♦r♥③t♦♥ strt ♦♦♣rt s♦r♥③ t♥t ♦ tt ♣♦st♦♥ ♥♦r♠t♦♥ t♦ ♠♣r♦ tr ♥② ♦♥sq♥t② t s♠♥t♦r② t♦ ♦♣ t ♦③t♦♥ ♦rt♠s t♥ ♥t♦ ♦♥t t s♣ts ♦s ♥t♦rs s ♣tr ♣rs♥ts t♦ tr♦♥♦s ♦③t♦♥ t♥qs s♦♥ ♦♥ ♥ ♦♥ ♥tr ♥②ss ♥ ♦♥ t ♦tr ♥ ♦♥ ②♣♦tss tst♥ Pr♦r t♦♦rt♠s sr♣t♦♥ stt ♦ t rt ♦♥ t ♦③t♦♥ strts ♥ t ①st♥t♥qs s ♣r♦ ♥ st♦♥

s♦♥ st♦♥ ♣rs♥ts ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ P♥ ♦r♥ ♦♠tr s ♠t♦ ♦r tr♦♥♦s ♦③t♦♥ ♠t♦ ss ♦♥ ♥tr ♥②ss ♣♣r♦s t♦ ♣r♦r♠ t ♣♦st♦♥ st♠t♦♥ r♥t♣r♦ss♥ st♣s r sr r♦♠ t ♦♠tr sr♣t♦♥ ♦ t Ps t♦ t ♣♦st♦♥st♠t♦♥ ♥② t ♦rt♠ s ♦♠♣r t♦ ①st♥ ♠t♦s s♥ s♠t♦♥s

❯s♥ ♣♦ssts ♦r ② ts ♦♠tr ♣♣r♦ t st st♦♥ ♣r♦s♥ ♦r♥ ♦rt♠ s ♦♥ ②♣♦tss tst♥ ♠t♦ ♣r♦♣♦ss t♦ ♠♥t♥ t♣♦st♦♥ st♠t♦♥ r② s♥ r♥ ♦srs ♥ ♣r♥t ts rt♦♥ ② ♣♦r ♦srs ♦r tt ♣r♣♦s t ♣♦r ♦srs r ♦♥② s t♦ t♥ t ♠♦s ♣♦st♦♥s rt ② r♥ ♦srs t ♠t♦s t ♥ s♠t♦♥ s♥ ♦♥t r♦ ♣♣r♦ ♥ tst ♦♥ ♥♦♥♦♦♣rt ♦③t♦♥ ♣r♦♠

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

s st♦♥ ♣r♦♣♦ss ♥ ♦r ♦ t ♦③t♦♥ t♥qs ♥♥ t r♥t ss ♦ t ♣♦st♦♥ ♥♦r♠t♦♥ ♥ t ♠♥ ♠t♦s ♦♥ ts st♠t♦♥ rst ♣rt srs ♦ t ♥♦ t ♣♦st♦♥ ♥♦r♠t♦♥ ♥ ♥t ♦t t♦ ♥ sr ♥ t♦ t ♥t♦r s♦♥ ♣rt ①♣♥s ♦ t♦ ♦t♥ t ♣♦st♦♥ ♥♦r♠t♦♥ ♥♦t② ② sr♥ t ①♣♦t ♦srs ♦ rss ♥t♦r ♥② ♥t②♥ tr r♥t ①♣♦tt♦♥ strts st ♣rts t r♥tt②♣s ♦ ♦rt♠s s ♦♥ ♥ r ♣♣r♦ ♥ ♦♥ ♦♠tr ♣♣r♦

♣♣t♦♥s ♦ ♦③t♦♥

❲t ♦t t ♥ t② t♦ ♦t♥ ♣rs ♣♦st♦♥ ♦♥ ♠♦ s ♥ t♥rs♥ ♣♦r ♦ t♦s s ❬❪ ♥ srs r ♥ rt ♠r♥ ♦ t♦s srs s ♥♦ ♣♦ss s t♦rs ♦ t ♦♠♠♥t♦♥ ♥ r t ♥ sr s s♠rt t♦ ♣r♦ ♣♦st♦♥ ♥♦r♠t♦♥ tt ♣ ♦ s♣ ♣♣t♦♥s t ♦♠♠♥t♦♥ ♥t♦r s t♦ ♦♥♥t ♠♦s t♦ ♥rstrtr ♥ ♥② t ♣r♦r ♠♥♥ t ♥rstrtr s t♦♦t t ♥ssr② t ♦r tt♥ t ♦t♦♥ ♥♦r♠t♦♥

s ♦ ts ♦③t♦♥ ♥♦r♠t♦♥ ♣♦t♥t② ♥ts ♦t t♦ t ♥ sr♥ t♦ t ♣r♦r r♦♠ t ♥ sr ♣♦♥t ♦ t ♣♦st♦♥ ♥♦r♠t♦♥ ssr ♣♣t♦♥s ♥ ♦ t♦s tt ♠♠t② ♦♠s ♥ ♠♥ s t ♥t♦♥♥ tr♥ ♣♣t♦♥ t♦♥ ♦♥ssts ♥ ♣r♦♥ rt♦♥s t♦ sr t♦ trt st ♥♦♥ ①♠♣ ♦ ts t♥♦♦② s ♣r♦② t tr♠♥ r♥t s ♦♥♦ ♥t♦♥ stt s②st♠ st ♥♦♥ s②st♠s r ♦ P♥ ♥ r♦♣ ❯ ♥ ss rs♣t② r♥ ♦s t♦ ♦♦t ♠♦♠♥t ♦ t sr ♥t♦ ♥ ♥r♦♥♠♥t ♦♥r♥ ♦ ♥t♦♥ ♥tr♥ ♣s ♣r♦♥ tr♥sr sr ♦t s ♥ ♦♦ ♣s t♦♥ ♣♦st♦♥ ♥♦r♠t♦♥ ♥ s♦ s s ♣rt ♦ t ♦♦③ ♥♦r♠t♦♥ ♥♥♦r♠t♦♥ ♦t t r♦♥ t② ♦r s♥ss ♥r♦♥♠♥t ♥ ♣t ♦r s♣②♦r♥ t♦ t rr♥t ♣♦st♦♥ ♦ t sr ② ①t♥s♦♥ ts ♥♦r♠t♦♥ ♠②s♦ rrr t♦ ♥ ♥ ♦r s♦♥ts t ♥r② s♦♣ ♦r t ♣♦st♦♥♥♦r♠t♦♥ ♥ s♦ s ♦r ♠r♥② srs s ♥ t ❯ t ♠♦♦♣rt♦rs t♦♠t② ♦t t ♣♦st♦♥ ♦ ♣♦♥ ♥ ♥ ♠r♥② sr❬❪

r♦♠ t ♥rstrtr ♥ ♥t♦r s t ♦t♦♥ ♥♦r♠t♦♥ r ♥r②s t♦ ♥♥ t qt② ♦ t ♥t♦r ❬❪

t♦ ♣rt ♥s ♥ t ♥r♦♥♠♥t ♥ ts ♣t s♥ s②♥r♦♥③t♦♥ t♦ ♣rt t ♦t♦♥ ♦ t ♥♥ s♥ t ♠♦♠♥t st♦r② t♦ ♣t tr♥s♠ss♦♥ s♣ts s t ♥t♥♥ ♦r♥tt♦♥ ♠♦r♠♥ ♦r tr♥s♠ss♦♥ ♣r♠trs ♠♦t♦♥ t②♣ t rt t♦ t rt ♣♦st♦♥ ♦ ♠♦ tr♠♥

♦r t♦s ♥♦r♠t♦♥ ♥ s♦ s ♥tr ② t ♥ sr ts ♥♦r ② t♥rstrtr t ② tr t♦r ♥trst ♥ ♦t♥ ♥♦r♠t♦♥ ❬❪ s ♥

tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

t s ♥ ♥r♦♥♠♥t ♠♦♥t♦r♥ r t ♣♦st♦♥ ♥♦r♠t♦♥ ♥ t♦ s♥s♦rs♦s t♦ rt ♠♣ strt♦♥ ♥ ts ①♣♥ t ss ♦ r♥t ♣♥♦♠♥s

t s ♥ s♦♥ tt ♦t♥♥ t ♦③t♦♥ s ♥ ♥trst ♦r ♠♥② ♣♣t♦♥s r♦♠ ♥ sr t♦ ♥rstrtr t r♥t ♠t♦s t♦ ♦t♥ tt ♣♦st♦♥♥♦r♠t♦♥ r♠♥s t♦ tr♠♥

t♥♥ P♦st♦♥ r♦♠ t ♦ srs

② ♥t♦♥ ♥ ♥♦ s rss r♦ t ♥ ♥♥♦♥ ♣♦st♦♥rs ♥ ♥♦r ♥♦ s r♦ ♥♦♥ ts ♣♦st♦♥ ♥ t ♦r ♠♥s♦♥♥ s♣ ♦③t♦♥ ♦♥ssts ♥t♦ ♦t♥♥ t ♣♦st♦♥ ♦ t ♥ ♥♦t t ♣ ♦ sr ♥♦r ♥♦s t t t♦♣ t ♦③t♦♥ ♣r♦ss ♥ s♥ s t♦ st♣s ♣r♦ss

rst② t ♥ ♥♦ ♦t♥s r♦ ♦srs r♦♠ t ♥♦r ♥♦s trr♦♠ t st♠t♦♥ ♦r r♦♠ t ♠sr♠♥t ♦ s♥ ♠trs

♦♥② ♦t♦♥ ♦rt♠ tr tt ♦srs ♥ tr② t♦ st♠t t♥ ♥♦ ♣♦st♦♥

♦♥ ♦ t ♦srs r♣rs♥t ♦t♦♥ ♣♥♥t Pr♠tr P ♥ s♥ s ♦♥str♥t s♥ t ♦♥str♥ts t ♣♦ss ♣♦st♦♥s ♦ t ♥ ♥♦

♥ s r t♦s Ps r ♠sr t♦t ♥② rr♦rs tr♥ ♦ tP ♦ ♥ r♦♥ ♥s ♦ t ♥ ♥♦ st♥s tr s ♥♦♦t♦♥ ♣♥♥t Pr♠tr P tt r♦♥ ♥rts ♥t♦ s♥ ♣♦♥t s t ♥ ♥♦ ♣♦st♦♥

♦r ♥ r ♣♣t♦♥ Ps s ♦sr t ♥ t rr♦r tr♠ ♥rr♦r tr♠ ♠② sr ss ♣♥♥ ♦♥ t ♦sr P s②♥r♦♥③t♦♥ s♦♥ ♦r tt ♣r♣♦s t s rqr t♦ ♦t♦♥ ♦rt♠s t♦ t ♥t♦ ♦♥t t♦s ♣rtrt♦♥s ♦s rr♦r tr♠s r t♥ ♥t♦ ♦♥srt♦♥ s r③t♦♥ ♦ r♥♦♠ r sttst ♣r♦♣rts ♦ t♦s r♥♦♠rs ♦♠s ts ♥ ♥♣t ♦ t ♦t♦♥ ♦rt♠ ♥ t ♥ t♦♦ ♥t♦♦♥srt♦♥ t ♦sr Ps t ♦rrt ♣r♦r ♦♥ tr ♥rs

♥r♣r♥t♥ s t♦

♥r ♣r♥t♥ s ♠t♦ ♠s t♦ ♣r♦ ♥ ♥♦ ♣♦st♦♥ st♠t♦♥ ②♦♠♣r♥ ♥ t ♠sr♠♥t ♦ Ps s t st ♦ Ps s r♦♠ ts t ♥♦♥ ♣♦st♦♥ ♥ ♥♦ ♣♦st♦♥ s st♠t ② ♥♥ ♥ tts t st ♦ Ps ♦s② ♠t t ♦sr Ps ❬❪

Prt② ♣r♦r t♦ t♦ st♠t ♣♦st♦♥ ♥r♣r♥t♥ ♠t♦s rqr♥ ♦♥ ♣s r st P ♦t r♦♠ r♥t ♥♦rs ♥♦s ♦r r♥t♣♦st♦♥s r st♦r ♥t♦ ts ♥ tt ♦t♦♥ ♦ Ps s ♥ ♦t♥t ♥r♣r♥t♥ ♥ tr♥ ♥t♦ ♥ ♦♥♥ ♣s ♥♥ t ♣♦st♦♥ st♠t♦♥r♥t strts ♥ s t♦ ♠♥♠③ t st♥ t♥ t Ps ♦srt♦♥t♦r ♥ t ts ❬ ❪

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

t ♥r♣r♥t♥ ♠t♦s ♣♦t♥t② ♥ rt ♣♦st♦♥ st♠t♦♥ ❬❪t② ♦♦s② s♦♠ rs rst t② r ♦♠♣t t♦ sts s ♦♥ ♠sr♠♥t ♠♣♥ s rqr t♦ s♠♣ r ♥♠r ♦ ♣r♥ ♣♦st♦♥s ♥ t r r t② s ♦♥② t ts rqrs r ♠♦♥t♦ s♣ ♥ ♦rr t♦ st♦r t ♥♦r♠t♦♥ ♥ ♥ t s♠ t♠ tt ♥♦r♠t♦♥ st♦ ss r② st ♥ ♦rr t♦ r t t♥② ♥ t ♣♦st♦♥ st♠t♦♥♥② t② r ♥♦t ① s ♥② ♠♦t♦♥ ♥ t ♥♦r ♣♦st♦♥s ♦r strtr ♠♦t♦♥ ♥ t ♥ ♦ ♠♦② t P s ♥ ts rqrs r♦♥strt♦♥ ♦ t ts

♥ s t♦

♦♥trr② t♦ t ♥r♣r♥t♥ s ♠t♦s t r♥ s ♠t♦s ♣r♦♣♦s t♦rt② s t P ♦srs t♦ ♣r♦r♠ ♣♦st♦♥ st♠t♦♥ tr ♠♦st sPs ♦r ♥ r♥ s ♦t♦♥ ♦rt♠s ❬❪ t ♥ t ♦♦♥

♥ tr♥t ♥t♦r

Base station 1

Base station 2

Base station 3

Blind node

P3

P2

P1

①♠♣ ♦ ♦③t♦♥ s ♦♥

♥t♦r ♦srs ♥ ♥t♦♥ ♦ t r ♣♦r r♦♠ r♦ t♦ ♥♦tr ♦srs ♥ s t♦ ♣r♦r♠ ♣♦st♦♥♥ ss♠♥♥ ♥r②♥ ♣t ♦ss ♠♦ s s♦♥ ♥ r t ♣t ♦ss ♠♦ ♦st♦ ♠ ♥ ss♦t♦♥ t♥ t ♣♦r tt♥t♦♥ ♥ st♥ ss♠♥ ttr ♥♦♥ t ♣♦st♦♥ ♦ t ♠ttr ts ♠tt ♣♦r ♥ t ♣r♠trs ♦ t♣t♦ss ♠♦ ♦sr ♠ts r r♦♥ ♥ rs♣ s♣rr♦♥ ♥ ♥ ♥♦ ♣♦st♦♥ s st♠t t t ♥trst♦♥ ♦ t♦s r♦♥s♥ ♦ t ♥ts ♦ t ♦sr s t ♥ ♦sr t♦t ♥tr①tr ♥tr ♥♦ ♦♠♠♥t♦♥ ♥♦r ♦♠♣① st♠t♦♥ ♣r♦sss

tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

♠ ♦ rr ♦

Base station 1

Base station 2

Base station 3

Blind node

r2

r1 r3

①♠♣ ♦ ♦③t♦♥ s ♦♥ ♦

♦ ♦srs s ♥♦r♠t♦♥ ♦ t t♠ s♣♥ t♦ t s♥ t♦ ♣r♦♣t t♥ t ♥ ♥♦ ♥ ♥ ♥♦r ss♠♥ tt s♥s ♣r♦♣ts t ts♣ ♦ t c ≈ 3×108m/s t s ♣♦ss t♦ tr♠♥ ♥ ss♦t r♥ t♥t ♥ ♥♦ ♥ t ♥♦r s t ♥ s ♦♥ r s ♥ ♦sr♠ts ♦♥str♥t s♣ s r ♥ ♦r s♣r ♥ ♥tr ♦♥ ♥♦rt rs q t♦ t r♥ ♦ ♦t♦♥ s ♦rt♠ tr sr ♦s♦♥str♥ts ♥ tr② t♦ ♥ t ♥trst♦♥ ♦ t rs ♦r s♣rs

Prt② t♦ ♠♥ ♠t♦s ①st t♦ ♦t♥ tt ♦sr ♦♥② r♥♥♥ t t♦② r♥♥ ♥ ♦♥② r♥♥ t ♦ s ♦t♥ ② ♠sr♥ tt♠ ♣s t♥ tr♥s♠tt s♥ ♥ ts r♣t♦♥ t t♥q rqrst s②♥r♦♥③t♦♥ ♦ ♦t r♦ s t ♥ ♦ t ♦♠♠♥t♦♥ ♥♥ ♥ t♦② r♥♥ ♠t♦ d0 tr♥s♠t s♥ s0 t♦ ♥♦tr d1 ♥ t d1 rtr♥s ♥♦tr s♥ s1 t♦ d0 ♥s tt s♦♥ s♥ s1 s ♠ tt♠ ♣s t♥ t r♣t♦♥ ♦ s0 ♥ t ♠ss♦♥ ♦ s1 ❯s♥ tt ♠t♦ ♥♦♠♦r s②♥r♦♥③t♦♥ r rqr t♥ t t♦ s s r t ♦st ♥♠r ♦ rqr ♦♠♠♥t♦♥s ♥ ♣r♦♠t ♥ tr♠s ♦ ♦r♥ ♥t♦rs t ♥♦s ♥st②

r ss ♠② r t ♦s ♦srs r② ♥ s ♦ ♦ ♦♥② r♥♥ t r② ♦ t ♠sr♠♥t rs ♦♥ tr② ♦ t s②♥r♦♥③t♦♥ ♦ t ♥♦s

s t s ♥ s♥ ♥ t ♣r♦♣t♦♥ ♥♥ ♣rs♥ts ♠t♣ts ♠♥s tt ♦r s♥ ♠tt s♥ sr ② ♥ tt♥t rs♦♥s rr ② t tr♥s♠ttr t ♥ strt ♣♥♦♠♥ ♥ ♣♣r ♦♥t r s♥ ♥ ♥ ♣rtr ♦♥ t rst ♣t s t s♦② rtt♦ t t r♥

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♥ s ♦ stt♦♥ t rt ♣t s ♦ t♥ t st♥ s st♠t t ♥♦tr ♣t s ♦♥sq♥ t ♦sr s

♠ r♥ ♦ rr ♦

Base station 1

Base station 2

Base station 3

Blind node

①♠♣ ♦ ♦③t♦♥ s ♦♥ ♦

♦ s ♠t♦ ss t r♥ ♦ ♦ ♦ s♥s t♥ srtr♥s♠ttr t♦ ♣♦st♦♥ t ♥ ♥♦ s s♦♥ ♦♥ r t r♦♥ ♦ t♥ s♣ r♥ ② tt ♦sr ♦♥ ②♣r♦ ♥ rs♣ ♥ ②♣r♦♦♥ ♦③t♦♥ ♦rt♠s s ♦♥ ♦s s sr ♦ t♦s ♦srs t♦st♠t t ♣♦st♦♥ s t ♥trst♦♥ ♣♦♥t ♦ t♦s ②♣r♦s ♥ ♦ t ♥t♦ ♦ ♦sr ♦♥ ♦ s t rqrs ss s②♥r♦♥③ ♥♦s ♦ ♠♦r rs♥ ♥♦t

♦ s ♦t♦♥ ♦rt♠s r ♠♦r s♥sts t♦ t ♥♦s t♦ t ♥♦s t ②♣r♦s ♥trst♦♥ ♥ ♥rts ♥ sr r♦♥s♥ t♦ ♥♦♥ ♦♥① ♣r♦♠ r qt r t♦ ♣♣r♦①♠t t ss st♠t♦r

trts t♦ ♠♣r♦ t ♦③t♦♥

s t r♦ ♦srs r ♥♦t ♥ssr② r t♦ r♥t ♣rtrt♦♥s ♦ t ♥r♦♥♠♥t t s rqr t♦ ♦rt ♠♥s♠s t♦ ♦t♥ t st ♣♦st♦♥ st♠t♦♥ tr tr qts ♦r tt ♣r♣♦s r♥t strts ♥ ♥t♦s② ♦♠♥ r ♦♥sr ♦s strts r s tr ♦♥ st♦♥ ♦ t ♠♦r ♣♣r♦♣rt P ♣♥♥ ♦♥ t ♦♥sr stt♦♥ ♦r ♦♥ t♥t♦r ♦r♥③t♦♥ ♥ t ♣t② t♦ ①♥ ♥♦r♠t♦♥ t♥ t r♥t♥♦s

tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

tr♦♥♦s ♦③t♦♥

tr♦♥♦s ♦③t♦♥ ♦♥ssts ♥ ♦t♥♥ ♦srs r♦♠ r♥t s♦rss ♦r ♦③t♦♥ ♣r♣♦s ♦r r♦ ♦srs t ♠♥s ♦t♥♥ ♥ t②♣♦ P r♦♠ r♥t t②♣ ♦ ♦ ss ♥♦♦②s s ♠t♣t♦♥ ♦t ♦srs s ♣rtr② ♥trst♥ ♥ ♦♥t①t r t tr♥ s t♦ ♥ ♥rs♦ ♦t t ♥♠r ♦ s ♥ t ♥♠r ♦ s ♠ ❬❪ ❱ ♥t t ♦ ♦③t♦♥ tt ♥rs ♦ ♦srs ♣s ♦③t♦♥ tr r♥ ♥♦srs t♦ s♦ ♥ ♥r♠♥s♦♥ ♣r♦♠ ♦r t♦ ♥rs t ♣♦st♦♥ r②② ♥ r♥♥②

②r ♦③t♦♥

♥♦tr s♦t♦♥ t♦ ♥rs t ♥♠r ♦ ♦srs ♥ t ♦③t♦♥ ♦rt♠ s t♦ s r♥t t②♣ ♦ P r♦♠ ♥ ❬❪ ♠r② t♦ t tr♦♥♦s ♦③t♦♥ t ②r ♦③t♦♥ ♦s t♦ ♠t♣② t ♥♠r ♦ ♦srst♦ ♥rs t r② ♦ t ♣♦st♦♥ st♠t♦♥

strt t♦r

♣♣♦s t♦ t ♥tr③ s♠ r t ♥rstrtr st♠t t ♣♦st♦♥t t ♥♦r♠t♦♥ r♦♠ t ♥♦s ♦ t ♥t♦r ❬❪ t strt ♥t♦r ts t ♣♦st♦♥ st♠t♦♥ t♦ t ♥♦s ♥ rr t♦ tr ♦srs ❬❪♥ t ♥♦r♠t♦♥ s ♠ ♥ t ♠♦ ♦♥sq♥t②r t ♦ ♦ t ♥rstrtr ② ♠t♥ t ♥♠r ♦♠♠♥t♦♥ ♥ ♦♥sr♥ t ♦③t♦♥ r② ❬❪

♦♦♣rt♦♥

Base Station 1

Base Station 2

Cell 2

Cell 1

Blind node

Anchor node 1

Anchor node 2

♥ ♥♦ ♦t♥s Ps ♦srs ♥ ♥ ♦ts ♦ 1 ♥ 2

P♥♥ t♦ t strt ♥t♦r s t ♥♦t♦♥ ♦ ♦♦♣rt♦♥ ♦♦♣rt♦♥ s♥ts t t② ♦ ♥♦s t♦ ①♥ ♥♦r♠t♦♥ t♦t rqst♥ t ♥rstrtr

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

❬❪ s s♦♥ ♥ r ♥♦s ♦♠♠♥t t ♦tr t♦ t t ♥♦r♠t♦♥rqr t♦ st♠t tr ♣♦st♦♥ ♦ ♠♥ ♣♣r♦s r ♥s t♦ ♥ ♥♦♦♦♣rt♦♥ t♦♣ ♥ ♠ss ♣ss♥

t♦♣ ♣♣r♦s s♣♣♦ss tt s♠ ♣rt ♦ t ♥♦s ♦ ♥t♦r ss t♦ tr ♣rs ♣♦st♦♥s t t ♣ ♦ P ♦r ♥st♥ ❬❪ ♦tr ♥♦s ♦♥② rt ♥♦r♠t♦♥ t♥ t♠ s s ♥tr♥♦ st♥s♦t♥ r♦♠ ♦r ♦ tr P ♥♦r♠t♦♥ r♦♠ t ♥♦s ♥t ♣rs ♥♦r♠t♦♥ ♦ s♦♠ ♦ t♦ ♦t♥ ♥ rt ♣♦st♦♥ st♠t♦♥ ♦ ♥♦s

ss ♣ss♥ ♣♣r♦ s ♥ ♥♦r ss ♣♣r♦ s ♦♥ r♣ ♠♦s st♦r r♣s ❬❪ ♥♦s ♦ t ♥t♦r ♦♥② ♥♦ tr ♥t ♣♦st♦♥ ②♣t tr ♠r♥ strt♦♥s ♦ ♣♦st♦♥s s♥ tr ♦ ♣r♦r ♥ t ♣♦ tr ♥♦rs ❬❪ ♥st ♦ s♠♣② ①♥♥ Ps ♦r r♥♥ tr ♣♦st♦♥st♠t♦♥ t ♦ P♦r ♥st② ♥t♦♥ P ♦r ♥t♦♥ s ①♥t♥ ♥♦s

♦rt♠s s ♦♥ r t♦

♦③t♦♥ ♦rt♠s r t Ps t ①sts r♥t ♣r♦ss♥ s♠st♦ ♦t♥ ♣♦st♦♥ r r ♣rs♥t t ♠♦st ♦♠♠♦♥s r s ♣r♦ss♥♠t♦s ♥ t r♠r♦ ♦r ♦♥ ♥ts t t♦rt r② ♥ rr ♦ t ♦srs r♥s

❲t st qr ❲ st♠t♦r

st sqr ♣r♦♠ s ♥ ♦♣t♠③t♦♥ ♣r♦♠ r t ♥♥♦♥ s ♣rt ♦t s♠ ♦ sqrs ♦ tr♠s ss♠♥ tt a ♥ b r ♣r♠tr t♦r ♥♥ ♦srt♦♥ t♦r rs♣t② ♥ X t ♦♣t♠③t♦♥ r t st sqr♣r♦♠ ♥ ♦r♠③ s

minX

f0(X) =

n∑

i=1

(aTi X− bi)2

ss♠♥ s♦♠ ♥r③t♦♥ t ♥②t s♦t♦♥ X ♥ rtt♥ ❬❪

X = (ATA)−1ATb

t st sqr s s♠r t♦ st sqr ♣r♦♠ ①♣t tt tt♦r ♦♥ t ♦srt♦♥ t♦r s t♦ s ②s

minX

f0(X) =

n∑

i=1

wi(aTi X− bi)2

r wi s t t ♦ ♦sr i ❯s② ♥ ♦③t♦♥ ♣r♦♠ t ♦♥srts r t r♥ ♦ ♠♥t ♦ t ♦srt♦♥ t♦r ② ♥♥ C =

tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

diag(w0, ..., wi) t s ♣♦ss t♦ rrt ♥ ♦rr t♦ t t t ❲ ♣r♦♠

X = (ATC−1A)−1ATC−1b

♠♥ ♥t ♦ t ❲ ♣♣r♦ s ts s♠♣t② ♥ ts rt ♦ ♦♠♣tt♦♥ ♦st ❬❪ ♦r t rs♦t♦♥ ♠t♦ rs ♦♥ ♥r③t♦♥ ♣r♦r♥ ♠tr① ♥rs♦♥ ♥ tr② ♥ s ♠tr① s ♦♥t♦♥ ♦ rsstt ♣rtr ♣r♦♠ s♦♠ ♥♥♠♥ts ♥ ♣r♦♣♦s ♥ ❬❪ s♣t ♦ tss♦t♦♥ st sqrs ♠t♦s ♦♥② t t ♥r ♣r♦♠ ♥ ♦③t♦♥ ♣r♦♠♦♥t①t ♥ s♣② ♥ ♥♦♦r tt ②♣♦tss ♦ ♥rt② ♥♥♦t r♥t♦r tt ♣r♣♦s ①♠♠ ♦♦ ♠t♦s r ♥r② ♣rrr

①♠♠ ♦♦ st♠t♦r

ss♠♥ tt ♥ ♦srt♦♥ t♦r b ♦ t tr ♣♦st♦♥ X ♦ ♥ ♥♦ sstrt t t ♦♦♥ ♥st② p(b|X) t s ♣♦ss t♦ ♥ t ss♦t♦♦

f(X, b) = p(b | X)

♠①♠♠ ♦♦ st♠t♦r ♠s t♦ ♥ ♦ X ♠①♠③f(X, b) ♦r ♦③t♦♥ ♣r♣♦s t ♠♥s tt t st♠t ♣♦st♦♥ X s ♦t♥♦r

X = maxX

f(X, b)

t rsr ♦ ♠①♠♠ ♦♥ t ♦ ♦♦ ♥t♦♥ ♥ t♠ ♦♥s♠♥ Prt② tt sr♥ s ♥r② ♦♠♣t t ♥ trt ♣r♦r s①♣tt♦♥♠①♠③t♦♥ ♦rt♠ ❬❪ ♦r t t♥ss ♦ ts trt♠t♦ s ♠t t♦ ♦♥① ♦♦ ♥t♦♥s ♥ s ♦ ♥♦♥ ♦♥①t② ♦ ①tr♠ ♥ ♣r♦♣♦s s t s♦t♦♥ ♦ t st♠t♦♥ ♣r♦♠ ♦♥① ♦♣t♠③t♦♥t♥qs ♥ s♦ s ❬❪ t tr ♦♠♣tt♦♥ t♠ t♦ t♦ t st♠t♦♥

r♠r♦ ♦r ♦♥

♥ st♠t♦♥ t♦r② ♥ sttsts t r♠r♦ ♦r ♦♥ s♥ts t ♦r ♦♥ ♦♥ t r♥ ♦ st♠t♦rs ♦ tr♠♥st ♣r♠trPrt② t stts tt t r♥ ♦ ♥② ♥s st♠t♦r ♦ r X s tst s s t ♥rs ♦ t sr ♥♦r♠t♦♥ J(X) r♥ tt t ♥② ♦ ♥ ♥s st♠t♦r s ♠sr ♦♥ ts ♣t② t♦ r ts ♦r ♦♥♥② s♦t♦♥ ♥ tt ♦♥ ♦ s♥t s ♥t sr ♥♦r♠t♦♥ s ② ♦ ♠sr♥ t ♠♦♥t ♦ ♥♦r♠t♦♥ ♦ X sr ♥♦r♠t♦♥s q t♦ t r♥ ♦ ♣rt rt t rs♣t t♦ t ♥♥♦♥ ♣r♠tr♦ t ♦rt♠ ♦ t ♦♦ ♥t♦♥ ❬❪ ♥ ts ♥ s ttr ♦ t ♥rs ♦ t sr ♥♦r♠t♦♥ ♠tr①

CRLB(X) = tr[

J((X))−1]

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♦rt♠s s ♦♥ ♥tr ♥②ss t♦s

♥tr ♥②ss ♣♣r♦ s♦ ss Ps t♦ ♦t♥ t ♣♦st♦♥ t ts ♣r♦ss♥rs r♦♠ r ♠t♦s ♥ ts ♣♣r♦ ♣r♦♣♦ss t♦ r♣rs♥t r♥♦♠r x s♥ ♥ ♥tr ♥♦t [x] ♦♥t♥♥ t s♣♣♦rt ♥t♦♥ ♦ ts P♥ ♥ s ♦ ③r♦ ♠♥ ss♥ r♥♦♠ r q t ♥ st♥r t♦♥ σq t s ♣♦ss t♦ ♥ ♦♥♥ ♥tr [Iq] t

[Iq] = [q − 3σq, q + 3σq]

r t t♦r ♥srs ♦♥♥ ♥tr ♦r q s ♦ s ♥ ①♣rss♦♥ ♣rs♥t s♦♠ ♥ts s♣② ♥ t ♦♥t①t ♦

♦③t♦♥ t ♦s t♦ sr t ♦r ♦ r♥♦♠ r ♥ r② ♦♠♣t ♦r♠s♥ ♦♥② t t♦ ♦♥rs ♦ t ♥tr

t ♦s t♦ ♣r♦r♠ s♠♣ ♦♠♣tt♦♥ ♦♥ ♥trs tr t♥ ♦r♥ t♦♠♣① ①♣rss♦♥s ♦ P

t ♥ ♥ ♥♦♥ ♥r ♣r♦♠s t s♠♣t♦♥ ♥s ♣r③t♦♥ ♦ ♥tr ♣r♦ss♥♦s② tt r♣rs♥tt♦♥ ♦sss ♥♦r♠t♦♥ ♥ rrs ♦ t ♦♠♣t P ♥

♣rtr t ♦rrs♣♦♥♥ ♣r♦t② ♦ ♥ r③t♦♥ ♥♥♦t ♥♦♥♥ ♦♥② t ♠ts ♦ t ①rs♦♥ ♦ t r♥♦♠ r♠♥ ♥♦tr ♣r♦♠ ♥s ♦ r ♦trs s♦♠ ♥♦♥sst♥ ♥ ♣♣r ♥ t ♥tr ♦ ♥♦t r♥t♦♥t♥♥ t s♦t♦♥

♥tr ♥②ss t♦ ♦r ♦③t♦♥

♥tr ♥②ss t♦r② s ♥ ♦♣ ♦t ② ♦♦r ❬❪ ♥ ② ♥s♥ ♥ts t♠ ❬❪ t s t♥ ②rs ♥ t rt♦♥ ♦ t ♥tr ♥②ss ♦r♥♥♦ ♥♦♥ s ♦♠♣t♥ ♦r ts ♠t♦ ♠rs r♦♠ ♦♥♥tt② ♦ ts♦r♥ ♦rt♦r②

♥tr ♥②ss s r♦② s ♦r r② r♥t ♥ ♦ ♣♣t♦♥s r♦♠ ♠♦♥♦ ♠s ♥ ♣rts rt♦rs ❬❪ t♦♥ ♦ ♥sr♥s ♣r♦♠s ❬❪ tr♦r♦♦t ♣r♦♠s ❬❪

♦♦t s ♣r♦② t r ♥tr ♥②ss s ♥ t ♠♦st ♦♥sr♦r ♣r♠trs st♠t♦♥ ❬❪ ♥ s♣② ♦r r♦♦t ♦③t♦♥ ❬ ❪

♣r♦♠ ♦ ♦③t♦♥ s r② st ♦r ♥ rss ② ♥tr ♥②ss ♣♣r♦s s t s ♥ s♥ ♦r t ♥rt♥t② ♦ t ♦srs s t♦ ♣♦st♦♥ st♠t♦♥ t ♥ r② s s t s ♥ ♥t♦♥r t t ♣♦st♦♥ ♦ t sr s ♥r② ♣t s r ♥tr ♦♥ tst♠t ♣♦st♦♥ ♥ r t rs r♣rs♥ts t ♣♦st♦♥ ♥rt♥t② ♥st ♦♥ ♥q ♣♦st♦♥ ♦③t♦♥ s ♦♥ ♥tr ♥②ss rtr♥s t ♥rt♥t②♥tr s ♠♦r r ♥ tr♠s ♦ t ♥♦ ♦ t ♣♦st♦♥ r♦♠ t♦rt♠ s♦♥

tt ♦ t rt ♦♥ ♦③t♦♥ ♥qs

♥ ♦tr ♥t ♦r ♦③t♦♥ s ts t② t♦ s② ♠r ♦srs ♦r② r♥t ♥trs ♦r ♥st♥ ♥tr ♥②ss s ♥ s ♥ ♥ ♥P ♣♣r♦ r ♥rt s♥s♦rs r s ♥ t♦♥ t ♦srs t♦♦♠♣♥st t r♥ ♥②♦♥ t ❬ ❪

tr ♥ ♥ tst t ♦ ♦srs ♦r rr ♣♣t♦♥s ❬❪ t♥tr ♥②ss s ♥ ♠♦r r♥t② ♥tr♦ ♥ ♦ ♦ s♥s♦r ♦rs ♦r tr♥ ♥♦s ♣♦st♦♥s ♠t♦ s ♣r♦ ts ♥trst ♦♠♣r t♦ ♣rt tr♥ ♣♣r♦ ❬❪ ♦♥sq♥t② t ♠♥♠♥t ♦ ♦trs ♦srss ♥ t♦ ♥rs ts r♦st♥ss ♦ ♣♦st♦♥♥ ❬❪ ♦r t ♣rtr s ♦♥♦♦r ♦③t♦♥ ♥tr ♥②ss s ♥♦t ♥ ② ①♣♦r ①♣t ♥ ❬❪ r ♦srs r s t♦ ♣r♦r♠ t ♣♦st♦♥ st♠t♦♥ ♥ ♥ s♣t♦ t ♣t② t♦ ♠r r♥t t②♣ ♦ ♦srs s ♥ ♥t ♥ ♦ttr♦♥♦s ♥t♦rs ♥ ②r t s♦♥ s♥r♦s t s ♥♦t tt ♥tr♥②ss ♠t♦s r st ♥♦t ♦♥sr ♦r ♦③t♦♥ ♥ r♦ ♦♠♠♥t♦♥s♦♥t①t

t r♥t ①♠♣s ♦ ♣♦st♦♥ st♠t♦♥ s ♦♥ ♥tr ♥②ss ♣rs♥t♥ ts st♦♥ r② ♦♥ ♦♥ ♦ ♦t ①st♥ ♠t♦s t♦ s♦ ♥ ♥tr ♥②ss ♣r♦♠

t ♦♥trt♦♥ s ♠t♦s t st♦♥ s ♠t♦s

♦♥trt♦♥ s ♠t♦s

♦♥trt♦♥ s ♠t♦s ♦♥sst ♥ trt② r♥ ♦♥trt♥ t ♥tr♦ t s♦t♦♥ ♥ rr ♦ ♦t t ♥♦♥ r② ♦ t ♥♣t ♣r♠tr ♥ tr♦rrs♣♦♥♥ t②♣ ♦ ♦♥str♥ts Prt② t ♠♥s tt t ♥tr ♦ r♣♥s ♦♥ t ♥trs ♦ t ♦tr rs ♥♦ ♥ t s♠ ♦♥str♥t tss♣♣♦s ♠♥♠ ①♠♣ t rs a b ♥ c s♣♣♦s t♦ ♥ ♥trs[a] = [20, 30] [b] = [−15, 25] ♥ [c] ∈ [−50, 10] rs♣t② ss♠♥ tt t②r ♥ ② t rt♦♥ c = a + b t s ♣♦ss t♦ s t ♥tr ♥②ss ♥ tst ♠t♠t rs sr ♥ ❬❪ t♦ ♦♠♣t

[c] = [a] + [b] = [20, 30] + [−15, 25] = [5, 55]

❯s♥ t ♣r♦r ♥♦r♠t♦♥ ♦♥ [c] t s ♣♦ss t♦ ♣t t ♥tr ♦ c

[c] = [c] ∩ [5, 55] = [−50, 10] ∩ [5, 55] = [5, 10]

♥tr r c s s ♥ ♦♥trt ♥♦tr s♣t ♦ t ♦♥trt♦♥ s♠t♦ s t ♦♥str♥t ♣r♦♣t♦♥ ♦ t rsts t ♠♥s tt t ♦♥trt♦♥♦♥ ♥tr c ♠② s♦ ♦♥sq♥s ♦ ♥trs a ♥ b ♦♥t♥♥ t s♠①♠♣ t ♦♥trt♦♥ ♦ [c] ♥ s♦ ♣s t♦ ♦♥trt [b]

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

[b] = ([c]− [a]) ∩ [b]

= ([5, 10]− [20, 30]) ∩ [b]

= ([−25,−10)] ∩ [−15, 25]

= [−15,−10]

t ♥♦t [a] ♥ ts ①♠♣

[a] = ([c]− [b]) ∩ [a]

= ([5, 10]− [−15, 25]) ∩ [a]

= ([−20, 35)] ∩ [20, 30]

= [20, 30]

♦r ♦♠♣①s ♣r♦♠s t ♠♦r rs ♥ ♥♦♥ ♥r qt♦♥s tt ♦♥str♥t ♣r♦♣t♦♥ ♣r♦ss ♥ r♣t t♦ ♥ ♥ ♦♣t♠ ♥tr ♦r ♣r♠trs t ♦♥trt♦♥ ♠t♦ ♥ r② ♥t ♥ tr♠s ♦ s♣ ♥ r② ♥♦③t♦♥ ♣r♦♠s ♦r t rqrs s♣ rr ♠♣♠♥tt♦♥ ♥ ♦rrt♦ ♦rrt② ♠♥ t ♥trs rt♠t

st♦♥ s ♠t♦s

❯♥ ♦♥trt♦♥ s ♠t♦s st♦♥ s ♠t♦s ♦ ♥♦t ♣♥ ♦♥ ♥tr rt♠t t ♦♥ rrs ♣r♦ss ❬❪ st♦♥ s ♠t♦s ♦♥ssts ♥s♣t♥ t ♣r♠tr ♥trs ♥ t♦ q ♥trs ♥ ♥ t t② ♦ ts ♥trs ♥ rrs ♦ t ♦♥str♥ts ♥trs r ♥♦t r r♠♦rs t ♦trs r ♥t ♦r rtr st♦♥ trt♦♥ ♦r ts ♦ ttst♦♥ ♠t♦ s ♣r♦ ♥ st♦♥ r♦♠ t ♦ ♠♣♠♥tt♦♥♣♦♥t ♦ st♦♥ ♠t♦s r r② ♦♥♥♥t s t② ♦♥② ♥♦ r②s ♦♣rt♦♥s ♥ t s♦♥ ♦♥ t t② s ♦♥ t s♠♣ ♦♠♣rs♦♥t♥ t ♦♥rs ♦ ♥ ♥tr ♥ t ♦♥str♥t ♦r tt rs♦♥ st♦♥s ♥ ♣rrr ♥ t ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ P ♣rs♥t♥ t ♥①t st♦♥

♥ t♦♥ t s♦ ①sts ♥tr ♥②ss rs♦t♦♥ ♠t♦ s♥ ♦♥t ♦♥trt♦♥♥ st♦♥ ♥ ♦ t ♠♦st ♠♦s s t ❱ ♦rt♠ sr ♥ ❬❪ s♠t♥♦s s ♦ t♦s ♠t♦ s ♣r♦ ts ♥② t ♦♥t ♦♥sr ♥ts ♦r ♦r t ♠♣♠♥tt♦♥ rs♦♥ ♠♥t♦♥ ♦r

sr♣t♦♥ ♦ t ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ P

♣r♦♣♦s P s ♥② ♦tr r♥ s ♦③t♦♥ ♠t♦s ss r♦♦srs t♦ st♠t ♣♦st♦♥ ♥st ♦ s♥ rt② t s ♦ t ♦srs


t P ♦♥rts t♠ ♥t♦ ♦♠tr s♣s ♥ sr tr ♥trst♦♥ ♥ t♥ s♣ t♦ rs♦ t ♣♦st♦♥♥ ♣r♦♠

♥ t ♦♦♥ t sr♣t♦♥ ♦ t ♦♥str♥ts ss♦t t t ♠♦st sr♦ ♦srs s ♣r♦ ♥ ♥ t ♦♠♣t ♣r♦ss♥ ♦ t P♦rt♠ s sr ♥ ♥② t ♠t♦ s t ♥ ♦♠♣r t♦ ♣♣r♦ ♥ s♠t♦♥

♦♥str♥ts ♥ ♦♠tr ♦♥str♥ts ♥t♦♥s

ss♠♥ ♥ ♠♦ ♦r r♦ ♦srs t s ♣♦ss t♦ ♦t♥ ♠t♠t ①♣rss♦♥ ♥s t rr ♥ tr♥s♠ttr ♣♦st♦♥s t ♠t♠t①♣rss♦♥ s s s ♦♥str♥t ♦ t ♣r♦♠

♥t♦♥ ♦♥str♥t ♦♥str♥t s ♠t♠t ①♣rss♦♥ rstrtst s♣ ♦ ♣♦ss s♦t♦♥s ♦r ♥ r

ss♦t t♦ ♦♥str♥t t ①sts ♦♠tr ♦♥str♥t ♠ts r♦♥ ♦ s♣ ♣♦♥ts ♦ t ♣♥ ♦♥♥ t♦ ts r♦♥ sts② t♦♥str♥t

♥t♦♥ ♦♠tr ♦♥str♥t ♦♠tr ♦♥str♥t s t ①♣rss♦♥♦ ♦♥str♥t ♥t♦ t ♥ s♣

♥ t ♦♦♥ t②♣s ♦ Ps r ♦♥sr t♦ t ♦♥str♥ts t r♥st r♥ ♦ r♥s ♥ t r ♣♦rs

♥ ♦♥str♥t

r♥ ♦sr r t t♥ ♥ ♥♦r t ♣♦st♦♥ a = [xa, ya, za] ♥t ♥ ♥♦ t ♣♦st♦♥ b = [xb, yb, zb] ♥ rtt♥

r = ‖a− b‖+ δr,

r δr s ♥ t♦♥ r♥ rr♦r

r♥ ♦ ♥s ♦♥str♥t

r♥ ♦ r♥s ∆ r♦♠ t♦ ♥♦rs t ♣♦st♦♥s a a′

♥ rtt♥

∆ = ‖a− b‖ − ‖a′ − b‖+ δ∆,

r δ∆ s ♥ t♦♥ r♥ ♦ r♥s rr♦r

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

P♦r ♦♥str♥t

♦ r ♣♦r ♦sr ♥ ♠♦ t st♥r ♣t ♦ss ♠♦

P (d) = P0 − 10np log10(d) +X,

r P0 s t ♣♦r r t ♠tr np s t ♣t ♦ss ①♣♦♥♥t ♥ X rr♦rr③t♦♥ s ♦r♥ t♦ ❬❪ t st♥ d ss♦t t♦ P t♥ ♥ ♥♦rt ♣♦st♦♥ a ♥ t ♥ ♥♦ t ♣♦st♦♥ b ♥ st♠t t

d = exp(M − S2) + δP

t

M =log(10)(P0 − P )

10np

S = − log(10)σX10np

r σ2X s t r♥ ♦ t r ♣♦r ♦srt♦♥ ♣rtrt♦♥ t♦♥δP st♠t st♥ rr♦r ♥ ♦♠♣t t

δP =√

(1− exp (−S2)) (exp (2M − 2S2))

Prt② t ♦ r ♣♦r ♥♦r♠t♦♥ ♥ ♦t♥ r♦♠ t rs♥ str♥t ♥t♦rs

♦♥♥ ♥tr tr♠♥t♦♥

t s ♥ s♥ tt t s ♦ t ♠t♠t ♦♥str♥ts ♣♥ ♦♥ t♦srt♦♥ ♦ ♥ rr♦r tr♠ ss♠♥ ♥ ♣r♦t② ♠♦ ♦r tt rr♦r t s♣♦ss t♦ ♦♥ ♦♥str♥t t ♦♥♥ ♥trs

♥t♦♥ ♦♥♥ ♥tr ♦♥♥ ♥tr ♦ ♥ ♦♥str♥t s ♦♥ ♥tr ♥s ♥② stss t ♦♥str♥t

♥ ss♠♥ tt t ♣r♦t② ♠♦s ♦s♥ ♦r δr δ∆ ♥ δP r ③r♦ ♠♥ss♥ t σ2r σ

2∆ ♥ σ2P tr r♥s rs♣t② t s ♣♦ss t♦ [Ir]

[I∆] ♥ [IP ] t ♦♥str♥t ♥tr ♦ t r♥ ♦ t r♥ ♦ r♥s ♦♥str♥t♥ ♦ t ♣♦r ♦♥str♥t rs♣t② s

[Ir] = [r − γσr, r + γσr]

[I∆] = [∆− γσ∆,∆+ γσ∆]

[IP ] = [P (d)− γσP , P (d) + γσP ]

t γ ♥ st♠♥t t♦r ❲t♦t ♣r♦r ♥♦r♠t♦♥ γ = 3 t♦ ♥sr ♦♥♥ ♥tr ♦r t ♦♥str♥t

s ♦♥sq♥


t ♦♠tr ♦♥str♥t r♦♠ r♥ ♦♥str♥t s s♣ s ♥ ♥♥s ♥ t♦♠♥s♦♥s ♦r s ♥ tr ♠♥s♦♥s t ♥tr a

t ♦♠tr ♦♥str♥t r♦♠ ♦ s s♣ s t r♦♥ t♥ t♦ ②♣r♦s ♥ ♦r t♦ ②♣r♦♦s ♥

t ♦♠tr ♦♥str♥t r♦♠ ♣♦r ♦♥str♥t s s♣ s ♥ ♥♥s ♥ t♦♠♥s♦♥s ♦r s ♥ tr ♠♥s♦♥s t ♥tr a

♦♠tr ♦rt♠ sr♣t♦♥

r♦♠ t sr♣t♦♥ ♦ t ♦♥♥ ♥tr ss♦t t♦ r♦ ♦srs ts ♣♦ss t♦ t ♦♠tr ♣♦st♦♥♥ ♦rt♠ ♠r② t♦ t r♣r♦♠ r t st♠t♦♥ ♦ t ♣♦st♦♥ ♥ ss♠t t♦ t rs♦t♦♥ ♦ s②st♠ ♦ ♦♥str♥ts t ♦♠tr rs♦t♦♥ ♦♥ssts ♥ ♥♥ t r♦♥ ♠t② t ♥trst♦♥ ♦ t ♦♠tr ♦♥str♥ts ♥ t♥ st♠t ♣♦st♦♥ ♥ ttr♦♥ P ♣r♦♣♦ss t♦ t ♣♦st♦♥ st♠t♦♥ ♣r♦♠ ♥ st♣s t♥ t ♦♦♥ ♥ s♠♠r③ ♥ t

Pr♦♣♦s ♦♠tr t♦ ♦rt♠ sr♣t♦♥

t ♦♥str♥ts r ♦① t ♦♥str♥ts r r t ♦♥str♥ts t♦ ♦t♥ ♠r ♦① r ♣♣r♦①♠t t ♠r ♦① t r ♦rt♠ t♦ ♦t♥ ♥ ♣♣r♦①♠t

r♦♥ r st♠t t ♣♦st♦♥ r♦♠ t ♣♣r♦①♠t r♦♥ r

t ♦♥str♥ts

rst st♣ ♥ P ♦♠♣t♥ s t♦ ♦♥rt t r♦ ♦srs t♦ ♦♥str♥ts r♦ ♦sr sr t♦ s t♦ ♣r♦r♠ t ♦③t♦♥ s ss♦t t♦ rt♥ ♦♥str♥t t②♣ s sr ♥ ss♠♥ t ♥♦ ♦ t ♠♦♣r♠trs t s ♣♦ss t♦ tr♠♥ t ♦♥♥ ♥tr ♦r t♦s ♦♥str♥tsr r♣rs♥ts t ♦♠tr ♦♥str♥ts r♦♠ tr ♦♥str♥ts ♦r ♥ ①♠♣ ♦ ♦③t♦♥ s♥r♦ t ♥♦rs r

♥♦rs ♣r♦ ♦ ♦sr ♥♦r ♣r♦s ♦ ♦sr ♥♦r ♣r♦s ♦sr

♦① t ♦♥str♥ts

♥st ♦ ♣rs② ♦♥♥ t ♥trst♦♥ r♦♥ ♦ t ♦♥str♥ts ♥ t♠ ♦♥s♠♥ ts t s rst ♣rrr t♦ ♥ t ♥♦s♥ ♦① ♦ ttr♦♥ ♦r tt ♣r♣♦s ♦♥str♥t ♥tr s ♣r♦t ♦♥ ①s ♥ ♦rr t♦

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

tr ♦♥str♥ts r t t♥s t♦ tr r♥t t②♣s ♦ r♦ ♦srs

tr ♦♥str♥ts r ♦①


♦t♥ ♦♥str♥t ♦① B s ♥ ♥ s ♦♥sq♥ t s ♣♦ss t♦♥ ♦♠tr ♦① r♦♠ ♦♥str♥t ♦① s r♣rs♥t ♥ r

♥t♦♥ ♦① ♥ ♥ ①s ♠♥s♦♥ k st t k = 2 ♦r ♣r♦♠ ♥♥♥ projk([I]) s t ♣r♦t♦♥ ♦ t ♥tr [I] ♦♥ ①s k ♦① B ♥ ♥t ❬❪

B = [I] = [I1]× [I2]× . . .× [Ik],

t [Ik] = projk([I]),

♥ rtt♥ t ♥ ♦sr tt ♦① trs s ♠ ♥tr s t ♥♠♥s♦♥ k

♥t♦♥ ♦♠tr ♦① ♦♠tr ♦① s t ①♣rss♦♥ ♦ ♦♥str♥t ♦①♥t♦ t ♥ s♣

♠rt r ♦ t ♦♥str♥ts

♦r t ♥② ♦ t ♥①t ♦♠♣tt♦♥ st♣ t ♠r ♦① rst♥ r♦♠ t♦♥str♥t ♦①s ♥trst♦♥ s t♦ t s♠st s ♣♦ss r♦r s♠rt♠r ♣r♦ss s ♣r♦♣♦s t♦ ♦♣t♠③ t s③ ♦ t ♠r ♦①

rst tt ♦♣rt♦♥ rqrs t♦ sr t ♠r ♦① Bm r♦♠ t ♦♥str♥ts♦①s Bn t

Bm =N⋂

n

Bn

r n s t ♥♠r ♦ ♦♥sr ♦①sr♦♠ tt ♣♦♥t t♦ stt♦♥s ♥ ♦♥sr tr Bm ①sts ♠♥s tt tr s r♦♦♠ t♦ r t s③ ♦ t ♠r♦①

♦r Bm ♦s ♥♦t ①sts ♠♥s t t ♦①s r ♥♦t ♥trst♥♥ ♦t ss t s rqr t♦ ♣t t ♦♥str♥t ♦①s tr t♦ r ♦r t♦ ♠①st Bm ♦♥sq♥t② ♠♦t♦♥ ♦ t ♦♥str♥t ♦①s s t♦ ♣♣

tr t ♦♥str♥t ♦①s r r t♦ r t s③ ♦ Bm ♦r t ♦♥str♥t ♦①s r ♠♦ss t♦ ♠ ♣♣r Bm

Prt② t s③ ♦ t ♦♥str♥t ♦①s ♥ ♥ ② ♠♦②♥ t ♦tr st♠♥t t♦r γ ♣r♦ss s r♣t ♥t γ s r ♠t s♠rt ♠r s t ♥ ♦rt♠ ♥ strt ② r

♣♣r♦①♠t t r ♦①

♣♣r♦①♠t♦♥ ♦ t ♠r ♦① s ♣r♦r♠ s♥ st♦♥ ♠t♦ s♦sr ♥ r ts s rrs ♣r♦r ♦♥sst♥ ♥ rrs② s♣tt♥ t ♠r ♦① t♦ ♣♣r♦①♠t t r s♣ rt ② t ♥trst♦♥ ♦ ♦♠tr ♦♥str♥ts t ♣r♦r rqrs st♣s

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♠♣ ♠r ♠rt ♠r

strt♦♥ ♦ t ♠r♥ st♣ t r② r r♣rs♥ts t ♠r ♦① rst ♦ t s♠♣ ♠r ♦ t tr ♦♠tr ♦①s ♦ t ♦♥str♥ts rst ♦ t s♠rt ♠r ♦♥ t tr ♦♠tr ♦①s ♦ t ♦♥str♥ts ♥rs③ t ♠r ♦① s s♠r

♦rt♠ ♠rt r ♦rt♠

γ ⊲ t♦ ♥sr ♦♥♥ ♥tr α > αmin ♦r γ γmin ♦

♦r n ♥ N ♦Bn =♣ts③(Bn, γ) ⊲ ♣t ♦①s s③s

♥ ♦r

Bm =N⋂

n

Bn ⊲ ♦♠♣t t ♦① ♥trst♦♥ ♦ ♦♥str♥ts

Bm == ∅ t♥γ = γ + α ⊲ ♥rs♥ t ♦♥♥ ♥trα = α ∗ 2

sγ = γ − α ⊲ r♥ t ♦♥♥ ♥trα = α/2

♥ ♥


rst qtr trt♦♥ ♦♥♠r ♦①

❩♦♦♠ ♦♥ s♦♥ qtr trt♦♥ ♦t ♦①s r t♦t ♥ rt

r qtr trt♦♥ ♥ ♣♣r♦①♠t r♦♥s ♦♠♣♦s ♦ str ♦♦①s

qtr ♣♣r♦①♠t♦♥ t♦ tr♠♥ t ♥♦s ♥ ♠♦s ♦①s

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♣♣②♥ r ♦rt♠ ❬❪ qtr ♦rt♠ ♥ ♦r ♦tr♦rt♠ ♥ ♦♥ t ♠r ♦①

st♥ t rst♥ ♦①s ♦♥ ② ♦♥ t ♦♥str♥ts tr♠♥♥ ♦①s t♦ ♣ ♥ rstrt t ♣r♦ss ♦♥ t♠

r ♦rt♠

r ♦rt♠ s ♣rtt♦♥♥ ♦rt♠ s♣ts s♥ ♦t ♥t♦ sr ♦♠♣♠♥tr② ♦♥s ♦r t ♣r♦♠ t ♥ t ♦rt♠ s ♣r♦r♠ ♦♥ t♠r ♦① r♦♠ t ♠r ♦① t r ♦rt♠ rtr♥s 2k sts ♦ ♦①s Br k s t s♣ ♠♥s♦♥ Prt② ♦r ♠♥s♦♥ ♦ ♦① ts ♥tr ss♣t ♥t♦ t♦ ♦♠♣♠♥tr② ♥trs

st t ♦①s

♥ t 2k ♦①s ♥ ♦t♥ t② t♦ tst ♥ rr ♦ t♦♥str♥ts ♦① s tst ♦♥ ② ♦♥ t ♦♥str♥ts t♦ tr♠♥

♦① s ♥ ♥♦s ♦① ♦① s ♠♦s ♦① s ♦t

♥t♦♥ ♥♦s ♦① ♦① s s ♥♦s t ♥rst rt① ♥ trtrst rt① r♦♠ t ♥tr ♦ ♦♠tr ♦♥str♥t r ♥s t ♦♥♥♥tr ♦ t ♦♥str♥t

♥t♦♥ ♠♦s ♦① ♦① s s ♠♦s t ♥rst rt① ♦r

t rtrst rt① r♦♠ t ♥tr ♦ ♦♠tr ♦♥str♥t r ♥s t ♦♥♥♥tr ♦ t ♦♥str♥t

♥t♦♥ t ♦① ♦① s s t t ♥rst rt① ♥ t rtrstrt① r♦♠ t ♥tr ♦ ♦♠tr ♦♥str♥t r ♦ts ♦ t ♦♥♥ ♥tr♦ t ♦♥str♥t

♥ t ①♠♣ ♣rs♥t ♥ r t ♦① ♥♠r s t s ♥ ♥♦s♦① t ♦①s ♥♠r ♥ r t s ♠♦s ♦①s t ♦①s ♥♠r ♥ r t s ♦t ♦①s

♥ ①t s♦t♦♥ t♦ tst t ♦♥♥ ♦ ♦① t♦ ♦♠tr ♦♥str♥t ♦rqr r②♥ t ♦♥♥ ♦ rt① ♦ t ♦① t♦ t ♦♥♥ ♥tr ♦t ♦♥str♥t ♥ rr ♦ t ♥♠r ♦ ♦①s t♦ tst ♥♦♥ ①t tr♥t ♠t♦s ♣r♦♣♦s s tr♥t ♠t♦ ♦♥ssts ♥ tst♥ ♦♥② rt①s ♦ t ♦① t♥rst rt① r♦♠ t ♥tr ♦ t ♦♠tr ♦♥str♥t ♥ t rtst ♦♥

①♠♣ ♣rs♥t ♥ r strts tt ♠t♦ ts ♥ ♦♥str♥t ♦t♥ r♦♠ r♥ ♦sr c ♦t ♥ O ♥ ♣rtr t ③r♦ ♠♥ss♥ t σc ts r♥ ts rst♥ ♦♥♥ ♥tr Ic ♥ rtt♥

[Ic] = [c− σc, c+ σc]


1 2

56

3

4

①♠♣ r♣rs♥t♥ ♦♠tr ♦♥str♥t ♥ ♦①s ♣♥♥ ♦tr rt ♣♦st♦♥ t♦ t ♦♥♥ ♥tr t ♦①s t ♥♦s♠♦s ♦r ♦t

♦♠tr ♦♥str♥t ♥ ♦① st♥s c − σc ♥ c + σc r tst♥s ♠t ② t ♦♥♥ ♥tr ♦ t ♦♥str♥t st♥s Vmin ♥Vmax r t st♥ r♦♠ t ♦♥str♥t ♥tr O t♦ t ♦sr rt① ♦ t ♦① ♥t rtr rt① ♦ t ♦① rs♣t②

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♥ t s ♣♦ss t♦ ♥ Vmin ♥ Vmax t st♥ r♦♠ t ♦♠tr ♦♥str♥t♥tr t♦ t ♥rst rt① ♦ t ♦① ♥ t rtr ♦ t ♦① rs♣t② ♥♦r ♥ ♦① ♥ ♥ ♦♥str♥t ♦r ②♣♦tss t♦ tst

h0 : Vmin < c− σch1 : Vmin < c+ σc

h2 : Vmax < c− σch3 : Vmax < c+ σc

Prt② t♦s ♦r ②♣♦tss rtr♥ ♦♦♥ s r st r ♦r srr♥ t s ♦ Dmin Dmax c ♥ σc ♥ t s ♣♦ss t♦ tr♠♥ ♦ ♥t♦r ♥t♦♥s Le(Bn) ♥ La(Bn) rtr♥♥ r ♦ ♦① Bn s♥♦s ♦r ♠♦s rs♣t② ss♠♥ s♦♠ ♦ s♠♣t♦♥s t♦s ♥t♦rs♥ rtt♥

Le(Bn) = h0h1h2h3

La(Bn) = h0h1h3 + h1h2h3 + h0h1h2

s♦♥ ♥ ♦ trt♦♥

❯s♥ t rsts ♦ t ♥t♦r ♥t♦♥s t ♥♦s ♦①s r st♦r ♥ t♠♦s ♦①s r ♥t ♦r ♥ r trt♦♥ t ♦tr ♦①s rrt ♦ ♣r♦ss s r♣t ♥t µe sts ♦ ♥♦s ♦①s Be r ♦t♥r♣♦♥ t ♣r♦ss s st♦♣♣ ♥♦ ♥♦s ♦①s r ♦t♥ t ♣r♦ss st♦♣s♦♥ ♥ ♥♠r µa sts ♦ ♠♦s ♦①s Ba ♦rt♠ srs t ♦♠♣t♣r♦r ♦ t ♠♣♠♥t r ♦rt♠

st♠t♥ t P♦st♦♥

①♠♣ ♦ P♦st♦♥ st♠t♦♥ ♥ t ♣♣r♦①♠t r♦♥

r♥s♦t r♦♠ ♠♦♥strt♦r ♦ t ♣♣r♦①♠t r♦♥

st♠t♦♥ ♦ t ♣♦st♦♥ ♦♥ t strt ①♠♣ s♥ t♠♣♠♥t ♦rt♠ r♥ s t tr ♣♦st♦♥ t s t st♠t♣♦st♦♥ ♥ r ♦①s r ♥♦s ♥ ♠♦s ♦①s rs♣t②

r strts t ♣♦st♦♥ st♠t♦♥ st♣ t st♣ ♦♥② ♦♥ssts ♥ st♠t♥ t tr ♣♦st♦♥ r♦♠ t ♥tr ♦ ♠ss ♦ t st ♦ ♥♦s♥ ♦①s Be


♦rt♠ ♥tr ♣♣r♦①♠t♦♥ ② ♦①s

⊲ ♥t③t♦♥ ♦ t st ♦ ♠♦s ♦①s t t ♠r ♦① BmBa = Bm⊲ ♥t③t♦♥ ♦ t st ♦ ♥♦s ♦①sBe = r(Be) ≤ µe ♦r (r(Be) + r(Ba)) ≤ µa ♦

♦r b ♥ Ba ♦Q = r(b) ⊲ ♥ ♠♦s ♦① r s ♣♣♦r q ♥ Q ♦

Le(q) == r t♥ ⊲ st ♥♦srBe = Be+ q

s La(q) == r t♥ ⊲ st ♠t②Ba = Ba+ q

♥ ♥ ♦r

♥ ♦r♥

♦r Ba Be = ∅ r s sr♥s♦t r t ♥♦s♥ ♦①s ♠♦s ♦①s tr ♥ st♠t ♥ ♥♦ ♣♦st♦♥s r s♣② s♥ t ♦rt♠s③t♦♥ t♦♦

♦t tt ♥ ss r t st ♦ ♥♦s♥ ♦①s Be r s♦♥ts t ♣♦st♦♥st♠t ♦ t ♥t ♦ ♥ ♥ st♠t♦♥ ♣r♦r s ♦♥ ②♣♦tsstst♥ ♠t♦ s sr ♥

t♦♥ ♦ t ♦st ♦♠tr P♦st♦♥♥ ♦rt♠P

s st♦♥ ts t ♣♦st♦♥♥ ♣r♦r♠♥s ♦ t ♣r♦♣♦s P ♥♦♠♣rs t ♥st r ♠t♦s

♠t♦♥ ♥r♦

s♥r♦ ♣rs♥t ♥ r s ♥ s ♥ ❬❪ t♦ t ②r ♥tr♦♥♦s ♥ ❲ s ♠t♦s ♦r tt rs♦♥ ts s♥r♦ s s t♦♣r♦ ♦♠♣rs♦♥ t♥ t ♣r♦♣♦s P ♠t♦ ♥ t♦s r ♦♥s ♥ts s♥r♦ t ♥ ♥♦ st♠ts ts ♣♦st♦♥ β t t ♣ ♦ ♥♦rs ♣r♦♥tr t②♣s ♦ r♦ ♦srs ♥♦rs t ♣♦st♦♥s AP ♣r♦ r ♣♦r♦srt♦♥s P t ♥♦rs t ♣♦st♦♥s AD ♣r♦ r♥ ♦ r♥s ♦srt♦♥s ∆ ♥ t ♥♦rs t ♣♦st♦♥s AR ♣r♦ r♥ ♦srt♦♥s r str t②♣s ♦ ♥♦rs r r♥ ♦♥ t s ♦ tr r♥t sqrs ts

AP ∈ HP AD ∈ HD AR ∈ HR

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

Pr♠trs stt♥s

Pr♠tr ❱

HP [−1, 1]× [−1, 1] km2

HD [−100, 100]× [−100, 100] m2

HR [−10, 10]× [−10, 10] m2

σr 2.97 mσ∆ 3.55 mσX 4.34 dBnp 2.64P0 −40 dB

t HR ⊂ HD ⊂ HP ♥ ♥♦ s ss♠ t♦ sr ts ♣♦st♦♥ ♥ HR

♥ ♥♦ t ♣♦st♦♥ β rs r♥ ♦srt♦♥s r r♦♠ ♥♦rs t♣♦st♦♥s AR r♥ ♦ r♥s ♦srt♦♥s ∆ r♦♠ ♥♦rs t ♣♦st♦♥s AD♥ r ♣♦r ♦srt♦♥s P r♦♠ ♥♦rs t ♣♦st♦♥s AP

♣r♦r♠♥ ♦ t ♣r♦♣♦s ♦♠tr ♠t♦ s ♦♠♣r t♦ ♣♣r♦①♠t♦♥ ♥ t♦ t r♠r♦ ♦r ♦♥ ♦♥t r♦ s♠t♦♥ s♦♥ t s♥r♦ sr ♥ r ♣♣r♦①♠t♦♥ ss rs♠♣① ♦♣t♠③r ♥t③ t t st sqr s♦t♦♥ ❲ ❬❪ t♠♥s♦♥ ♦♦ ♥t♦♥s ♦rrs♣♦♥♥ t♦ t ♥ s♥r♦s r sr ♥❬❪ ♣rtrt♦♥ ♦ t ♦srt♦♥ ♦ t r♥ ♦♥str♥t t r♥ ♦ r♥♦♥str♥t ♥ t ♣♦r ♦♥str♥t r s♣♣♦s ③r♦ ♠♥ ss♥ t tr rs♣t r♥s σ2r σ

2∆ ♥ σ2P ♣r♠tr stt♥s ♥ ♥ ♦s♥

♦♠♣♥t t t ❲ ♠sr♠♥t ♠♣♥ ❬❪


♦♠♣rs♦♥ ♦ Pr♦r♠♥s

0 2 4 6 8 10

ˆ

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

P♦rs ♦ r♥s

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

P♦rs r♥s

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

♥s ♦ r♥s

0 2 4 6 8 10

0.0

0.2

0.4

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0.8

1.0CRLB

Geometric

ML-WLS

②r

s ♦ ♣♦st♦♥♥ rr♦r s♥ t ♣r♦♣♦s ♦♠tr ♠t♦ ❲♥ ♣♣ ♦♥ ②r ♣♦st♦♥♥ t♥q

tr ♦rt♠s r ♦♠♣r ♥ tr♠s ♦ ♠t ♥st② ♥t♦♥ r♦♦t ♠♥ sqr rr♦r ♥ ♦♠♣tt♦♥ s♣ ♦r t ②r ss ♦r ♦r②r ♦♥rt♦♥s

P♦rs r♥ ♦ r♥s s♥ r ♣♦rs ♥ r♥ ♦ r♥s♦srs r

P♦rs r♥s s♥ r ♣♦rs ♥ r♥s ♦srs r ♥s r♥ ♦ r♥s s♥ r♥s ♥ r♥ ♦ r♥s ♦srsr

②r s♥ r ♣♦rs r♥s ♥ r♥ ♦ r♥s ♦srsr

♥♦♥②r ss s♥ ♥q t②♣ ♦ ♦srt♦♥ ♦♥② r♥ r♥ ♦r♥s ♦r r ♣♦r r ♥♦t ♦♥sr r r♦♠ t ♠♣r s♦♥ ♥

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

r t ♣♣rs tt t ♣r♦♣♦s ♦♠tr ♠t♦ ♣rs ♦♥ ❲t ♥rs ♦ r② ♦ ♣♦st♦♥♥ s s♣② s♥♥t ♦♥ r ♥r ♦s t♦ ss s♥ r ♣♦r ♦srs ♦ ♥ t♥m♥ m ♦r ♥ ♥♦s tr ss s ♦♥② ♦♥ t♠ s ♦srs ss♦♥ ♥ r ♦r s♥ t②♣ ♦ ♦srs s s♦♥ ♥ r s♦s♦ ttr r② ♥ tr♠s ♦ ♣♦st♦♥ st♠t♦♥ ♦s rsts r ♦♥r♠ ②t s s♦♥ ♥ ♠♦st s♥♥t ♠♣r♦♠♥t s ♦sr♦r t ②r s♠ ♠①♥ ♣♦rs ♥ r♥s ♥ r t ♣r♦♣♦s ♦♠tr♠t♦ ♥srs cm ♥rs ♦ ♣♦st♦♥♥ r② ♦r ♥ ♥♦s r♥ ♥ 20× 20 m2 r♦♦♠

♦s② ts ♠♣r♦♠♥ts ♦♠ ♦t t t ♦st ♦ ①tr ♦♠♣tt♦♥ ♦♠♣①t② ♥ s♣t ♦ ♣r♦♥ ♦♠♣t ♦♠♣①t② st② r s♦s s♦♠♣r♠♥r② rsts s ♦♥ ♥ r ♦ ♦♠♣tt♦♥ s♣ ♦r ♠t♦ ♥t♦s st♦r♠s t ♥ ♦sr tt t ♣r♦♣♦s ♠t♦ s ♥r② s♦rt♥ t ❲ ①♣t ♥ t r ♣♦r ♦srs ♥ r♥ ♦srs r s t s♦ s♦s tt t r♥ ♦ r♥s ♦♥str♥t s t ♦rst ♥tr♠ ♦ s♣ rtr ♥stt♦♥ ♦ t♦ ♠♣r♦ t s♣ ♦ t r♥♦ r♥s ♦♥str♥t ♦r♦r t ♦♠♣rs♦♥ t♥ ♦t ♠t♦s s ♥r st ❲ ♥♠r ♦♣t♠③t♦♥ s s ♦♥ ♥ ♦♣t♠③ ♦♠♣ ♦rtr♥ ♦rs t ♣r♦♣♦s ♦♠tr ♠t♦ s s ♦♥ ♥♦♥ ② ♦♣t♠③ ♥tr♣rt♦ ♥ P②t♦♥ ♦♥sr♥ tt r♥ ♦ ♠♣♠♥tt♦♥ ♦♠tr ♠t♦s st s t ❲ ♦ s ♦r♦r t ♦♠tr ♠t♦ s ②♣r ♥ ♥♦s ♦♥② ♠♥tr② ♦♣rt♦♥s ♥ ♦ ♣r♦② r② ♥t②♠♣♠♥t ♥ t rr

Power+ ranges

Power +diff. of ranges

ranges +diff. of ranges

Full Hybrid0.00

0.05

0.10

0.15

0.20

0.25

tim

e(s

)

Geometric

ML-WLS

♣ ♦♠♣tt♦♥ ♦♠♣rs♦♥ t♥ t ♣r♦♣♦s ♦♠tr ♠t♦ ♥ ❲ s♥ ♥♠r ♦♣t♠③r

tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

s t♦

②r ♠♦ ♦♠tr ❲ m m m

P♦r ♦ r♥s P♦r r♥s ♥s ♦ r♥s ②r

tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

♥ ♦ t ♥ts ♦ t ♣r♦♣♦s P ♦rt♠ s ts ♣t② t♦ ②s♠t r♦♥ ♦ s♣ ♦♥t♥♥ t s♦t ♣♦st♦♥ ♥ ♥ stt♦♥ rt ♥♠r ♦ ♦srs s ♥♦t s♥t t♦ ♦t♥ t ①t ♣♦st♦♥ ♦ ♥♦P s t♦ ♠t ♦♥ ♦r sr r♦♥ ♦ s♣ ♣♦t♥t② ♦♥t♥♥ t ♥♦t ♣r♦♣rt② s s ♥ ts st♦♥ t♦ sr ♥ ♦r♥ tr♦♥♦s ♦③t♦♥♦rt♠

sr♣t♦♥ ♦ t Pr♦♠

♦♥sr s♥r♦ s strt ♥ r ♣♦st♦♥ B ♦ ♥ ♥♦s st♠t r♦♠ t t♦ ♥♦s② r♥ ♦srt♦♥s r1 r2 ♣r♦ ② t r♥ ♥♦st ♥♦♥ ♣♦st♦♥s R1 ♥ R2

ri = ‖Ri −B‖+ δi, i = 1, 2,

r δi s t rr♦r ♥ t r♥ st♠t ♥ ♣r♦t② ♠♦ ♦r δi t s♣♦ss t♦ tr♠♥ ♦♥♥ ♥tr ♦r t r♥ st♠t ri s s♦♥♥ r ②s ♦♥♥ r♦♥ s♣ s ♥ ♥♥s t ♥tr Ri ♠t② ♣r♦♠ ♦rs ♥ t ♥trst♦♥ ♦ t♦ ♦♥♥ ♥♥ s♣ts ♥t♦ t♦s♦♥♥t ssts C1 ♥ C2 t ♥tr♦s C1 ♥ C2 rs♣t② t♦ ssts♥ tr ♥tr♦s ♥ ♦t♥ s♥ t P ♠t♦ sr ♥

♦ ♣r♦ ♥ ♥♠♦s ♣♦st♦♥ st♠t t♦♥ ♥♦r♠t♦♥ s rqrt s ss♠ tt t♦♥ ♦srt♦♥s Pk ♦ t ♦ ♣♦r r r♦♠ t♣♥ ♥♦s t ♣♦st♦♥s Hk ♦ ♣♦r ♦srt♦♥ Pk s ♠♦ s st♥♣♥♥t ♠♥ µ(dk) st♦rt ② ♥ t rr♦r tr♠ Xk

Pk = µ(dk) +Xk,

t dk = ‖B − Hk‖ t s ss♠ tt t ♦♥t ♣r♦t② ♥st② ♥t♦♥ ♣♦ X1, . . . , XK r ♥♦♥ ♦ ♣♦r ♦srt♦♥s r tr ♥t♦ t♦rP = [P1, . . . , PK ] Prt② t ♦ ♣♦r ♥♦r♠t♦♥ ♥ ♦t♥ r♦♠ tr s♥ str♥t ♥t♦rs

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♥ ♥♦ t ♣♦st♦♥ B rs t♦ r♥ st♠ts r1 ♥ r2 r♦♠r♥♥ ♥♦s t ♣♦st♦♥ R1 ♥ R2 rs♣t② rr♦rs ♥ r♥ st♠ts r ♠♦② t♦ ♦♥♥ ♥♥ ♥tr t R1 ♥ R2 rs♣t② ♥trst♦♥ ♦ ♥♥r t t♦ s r♦♥s C1 ♥ C2 t tr ♥tr♦s C1 ♥ C2 rs♣t② ♥t♦♥ t ♥ ♥♦ s ♦ ♣♦r ♦srt♦♥ r♦♠ ♣♥ ♥♦ t♣♦st♦♥ Hk st♥ r♦♠ Hk t♦ Ci s ♥♦t ② dk,i

tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

Pr♦♣♦s s♦♥

♦♥t♦♥ ♣ s ♣♣r♦①♠t ♦r t ♣♦r ♦srt♦♥ ♦♥t♦♥ ② t♣♦st♦♥ ♦ t ♥ ♥♦ s

fP|B,r1,r2(p) ≈

fP|C1,r1,r2(p), B ∈ C1

fP|C2,r1,r2(p), B ∈ C2

0, ♦trs,

t p ∈ RK ♥ t ♣r♦t② ♦ t ♥t B /∈ C1 ∪ C2 s ♥t s

♣♣r♦①♠t♦♥ s ② ♣♣r♦♣rt② ♦♦s♥ t r♦♥s C1 ♦r C2 ♥ B s♥♥♦♥ fP|B,r1,r2(p) ♥♥♦t ♦♠♣t ♦r rt r♥ st♠ts r t♥ B ♥ ♣♣r♦①♠t ② C1 B ∈ C1 ♦r ② C2 B ∈ C2 ♥ ts

dk ≈

dk,1 =‖C1 −Hk‖, B ∈ C1

dk,2 =‖C2 −Hk‖, B ∈ C2.

❲t t ♦ ♣♣r♦①♠t♦♥s t s♦t♦♥ ♦ t ♠t② ♣r♦♠ ♥ ♣rss ss s♦♥ ♣r♦♠ r λ s t ♦♦ rt♦ s♦♥ trs♦γ ♥ ♥ t♦ ♦♥t t ♣r♦r ♥♦r♠t♦♥ ♦r ♦sts ❬❪

λ =fP|C1,r1,r2(p)

fP|C2,r1,r2(p)

C2

≶C1

γ.

s♦♥ r s ♦t♥ ♦r γ = 1

♣ s ❯♥♦rrt ss♥ ♦ P♦r rr♦rs

♥ t s♣ s r X1, . . . , XK r ♥♣♥♥t ss♥ r♥♦♠ rst ③r♦ ♠♥ ♥ r♥s σ21, . . . , σ

2K ②s ♦r B ♥ Ci

fP|Ci,r1,r2(p) =K∏

k=1

1√2πσk

exp

(

−(pk − µk,i)22σ2k

)

, i = 1, 2,

t µk,i = µ(dk,i) ♥ t ♦ ♦♦ rt♦ Λ = lnλ rs

Λ =

K∑

k=1

[

(pk − µk,2)22σ2k

− (pk − µk,1)22σ2k

]

s♦♥ r s ♦t♥ ♣♦♥ ♥srt♦♥ ♦ ♥t♦ t γ = 1

K∑

k=1

1

2σ2k

[

µ2k,2 − µ2k,1]

C1

≶C2

K∑

k=1

1

σkPk(µk,1 − µk,2).

t ♥ ♦sr tt ♦r ① ♥tr♦s C1 ♥ C2 t t ♥ tr♠s r ♦♥st♥ts t rt ♥ tr♠s r ss♥ r♥♦♠ r s t ♦♠♣tt♦♥♦ rr♦r ♣r♦t② s ♥♦♥ ❬❪ t♦♥ ♦ t ♣r♦r♠♥s ♦ ts♣rtr ♠t♦ r ♥ ♥ st♦♥

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

Pr♠trs stt♥s

Pr♠tr ❱

S [−20, 20]× [−20, 20] m2

L [−80, 80]× [−80, 80] m2

σδ 0.9 mσX 4 dBnp 3P0 −40 dB

r ♣♥ ♥♦s ♥ ♣♥ ♥♦s

♦♠♣rs♦♥ ♦ t s♦t ♣♦st♦♥♥ rr♦r t♥ t r♠r ♦♦r ♦♥ ❬❪ t st♠t♦r t trt st♠t♦r ♥t③ t t st sqrs s♦t♦♥ ❲ ♥ t ♣r♦♣♦s ♠t♦ t r ♣♥ ♥♦s ♥♦s ♥ ♥ ♣♥ ♥♦s

t♦♥ ♦ tr♦♥♦s ♦③t♦♥ s♥ ②♣♦tss st♥

♥ ts st♦♥ t ♣r♦r♠♥ ♦ t ♣r♦♣♦s ♠t♦ s ♦♠♣r t♦ ♣♣r♦s ♦♥t r♦ s♠t♦♥s ♦ t s♥r♦ sr ♥ r ♥♣r♦r♠♥s tr st♠t♦r s ♦♥sr r②♥ ♦♥ ♦ ♦♣t♠③t♦♥ ♥ ♥ ♣♣r♦①♠t♦♥ ❲ ♥ ♦ ♦♣t♠③r s ♥t③ t tst sqrs s♦t♦♥ ❬❪ ♦t ♥tr♦ ♥ st♦♥

♠t♦♥s ♥r♦

♣r♦r♠♥ ♦ t ♣r♦♣♦s s♦t♦♥ s ssss ♦♥t r♦ s♠t♦♥s s♣ s sr ♥ st♦♥ s ♦♥sr ♦r t st♣ ♥ ♥ r t t ♣r♠trs stt♥s s ♥ ♣♦st♦♥s B,R1, R2 r r♥♥♣♥♥t② ♥ ♥♦r♠② ♦♥ t r S ♣♦st♦♥s Hk ♦ t ♣♥ ♥♦sr ♥♣♥♥t② r♥ ♦r♥ t♦ ♥ ♥♦r♠ strt♦♥ ♦♥ t rr r L

tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

r♥ rr♦rs δ1 ♥ δ2 ss♠ t♦ ♥♣♥♥t ③r♦ ♠♥ ss♥ r♥♦♠ rst r♥ σ2δi = σ2δ ♠♥ ♦ t r ♦ ♣♦r s ♠♦ ♦r♥ t♦t st♥r ♣t ♦ss ♠♦ µ(dk)

µ(dk) = P0 − 10np log10(dk),

r P0 s t ♣♦r r t ♠tr ♥ np s t ♣t ♦ss ①♣♦♥♥t r♥σ2k s ♦s♥ q t♦ σ2X ♦r k ❱s ♦r P0 np σ2δ ♥ σ2X r ♦s♥ ♦r♥t♦ t ♠sr♠♥ts r♣♦rt ♥ ❬❪

①♠♠ ♦♦ st♠t♦♥

st♠t♦r ♦r t ②r ♣♦st♦♥♥ ♣r♦♠ rsB ∈ argmaxz Λ②r(z)r Λ②r(z) ♥♦ts t ♦ ♦♦ ♥t♦♥ ♦r B s ♦♥ ri ♥ P ♥♣♣r♦ s t♦ ♥ t ♠①♠ ♦ t ♦ ①tr♠ ♦ t ♦ ♦♦ ♥t♦♥♦t♥ ② qt♥ t r♥t t♦ ③r♦ ♦r ♥♣♥♥t r♥ st♠ts ♥ ♣♦r♠sr♠♥ts t r♥t ♦ t ♦ ♦♦ ♥t♦♥ rs

∇Λ②r(z) = ∇ΛP♦r(z) +∇Λ♥(z),

r ΛP♦r ♥ Λ♥ r t ♦ ♦♦ ♥t♦♥s ♦ t ♣♦r ♠sr♠♥ts♥ ♦ t r♥ st♠ts rs♣t② t♦ r♥ts r ❬❪ ❬❪

∇ΛP♦r(z) =K∑

k=1

1

s2Mk − s2 − ln ‖z −Hk‖

dk2 (z −Hk),

∇Λ♥(z) =

2∑

i=1

1

σ2τ

ri − ‖z −Ri‖ri

(z −Ri),

t t ♥t♦♥s

s = −σX ln 10

10np, Mk =

(P0 − Pk) ln 1010np

+ ln d0.

t♦ t ♥♦♥♥r rt♦♥ ♥♥ t r♦♦ts ♦ rqrs ♦ ♥♠r♦♣t♠③t♦♥ s ♥♦t s ♦r ♠♦st ♣♣t♦♥s ♦r ♥ ♣♣r♦①♠ts♦t♦♥ ♥ ♦t♥ ② ♥t③♥ ♥♠r ♦ ♦♣t♠③r t ♥ ♥t ss t st sqrs ❲ ♣♣r♦ ❬❪

♦♠♣rs♦♥ ♦ Pr♦r♠♥

♣r♦r♠♥s ♦ t tr ♦rt♠s r ♦♠♣r ♥ tr♠ ♦ ♠t ♥st②♥t♦♥s ♦tr rts ♥ r♦♦t ♠♥ sqr rr♦rs r♦♠ t ♠♣rs s♦♥ ♥ r t ♣♣rs tt ♦r ♦ ♥♠r ♦ ♣♥ ♥♦s t♣r♦r♠♥ ♦ t ♣r♦♣♦s ♠t♦ ♦t♣r♦r♠s ❲ ♥ s ♦s t♦ tt ♦ ♦r ♥♠r ♦ ♣♥ ♥♦s t ♣r♦♣♦s ♠t♦ ♥ ❲ s s♠r♣r♦r♠♥s ①♣t ♥ r rr♦rs r♠ r t ♣r♦♣♦s ♠t♦ ♣rs

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

s r rr♦rs r ♦sr t♦ ss rq♥t ♥ t ♥♠r ♦ ♣♥ ♥♦ss ♦ ♥s♣t ts r♥ t ♦rr♥ ♦ ♦trs s ♦♥sr ♥ ♦trs ♥ s ♦♦s B ∈ Ci t st♠t ♦ B s ♥ ♦tr t s ♥t ♦♠♣♠♥t ♦ Ci ♦t tt ♦r t ♣r♦♣♦s ♠t♦ ♥ ♦tr s q♥tt♦ s♦♥ rr♦r ♥ ♥ r t ♥ ♦sr tt t ♦trrt rss t t ♥♠r ♦ ♣♥ ♥♦s ♥rss ♦t sr♣rs♥② t st♠t♦r ②s t ♦st ♦tr rt ♦ t tr ♠t♦s t s♦ ♣♣rs tt t♣r♦♣♦s ♠t♦ ♦♥sst♥t② ♦t♣r♦r♠s t ❲ ♥ tr♠s ♦ ♦tr rt ss ♠♦st s♥♥t ♥ t ♥♠r ♦ ♣♥ ♥♦s s ss t♥ ♦r s r♥s♦ ♣r♦r♠♥ r s♦ rt ♥ t s r♣♦rt ♥ r ♥ ♣rtr♦r ♦r ♦r ss ♣♥ ♥♦s t ♥ ♦sr tt ♣r♦♣♦s ♠t♦ s ♦s t♦ t r ♦♠♣r t♦ ❲ r

tr rt s ♥♠r ♦ ♣♥ ♥♦s s ♥♠r ♦ ♣♥ ♥♦s

t♦♥ ♦ t ♥♥ ♦ t ♥♠r ♦ ♣♥ ♥♦ ♦♥ t trrt ♥

♦ ♦srt♦♥s sst tt t ♦r t ♣r♦♣♦s ♠t♦ ♣♣r♦♥ ttrt t♦ t♦ t②♣s ♦ rr♦rs r rr♦rs ♦trs t♦ s♦♥ rr♦r ♥ ♥ s♠ rr♦rs rst♥ r♦♠ t ♣♣r♦①♠t♦♥ ♥ s♠ rr♦rs♦r s♥ t ♥tr♦s C1 ♦r C2 r s t♦ st♠t t ♣♦st♦♥ ♦ t ♥ ♥♦s t s ♦♥tr tt t t ♦ ts s♠ rr♦rs ♥ r ② ♠♣r♦♥ts ♣♣r♦①♠t♦♥ ♦♥sr♥ t♦♥ ♥♦ ♦ t ♣r♦t② ♠♦ ♦r tr♥ rr♦r δi s ♥♦r♠t♦♥ ♦ ♥ rt② s t♥ ♥t♦♥ ♥t ♦♠♣tt♦♥ ♦ ♥tr♦s tr♥t② t ♣r♦♣♦s ♠t♦ ♦ s t♦♣r♦ ♥ ♥t ss ♦r ♥♠r ♦♣t♠③t♦♥ ♦ t ♦♦ ♥t♦♥ t srtr ♦♥tr tt t ♦tr rt s q♥t t♦ t rt ♦ s s♦♥♥ ♦ s♦ r ② ♠♣r♦♥ t ♣♣r♦①♠t♦♥

②♣♦tss st♥ t♦ s ①st♥ t♦s

s st♦♥ ♣r♦♣♦ss ♥ t♦♥ ♦ t ②♣♦tss tst♥ ♠t♦ ♦♥ ♥♦♥♣r♦♠ ♦ ♠ss ♣ss♥ sr ♥❬❪ ♦r tt ♣r♣♦s t s♥r♦ s

tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

trt ♥ r s ♦♥sr ♥ tt ♥ s♥r♦ ♥♦rs r ♥ ♥ ♥♦s tr② t♦ st♠t tr ♣♦st♦♥s ♥ ♥♦ s ♥ st② t ♦♥② ♥♦rs ♥♦s r♥t ♦r ♥ ♥♦

♥ tt stt♦♥ t ♣♦st♦♥ st♠t♦♥ ♦ ♥ ♥♦ s t♦ ♥ ♠t②♥ ♣r♦ss♥ t♦ r♦ ♦srs r ♥♦t s♥t t♦ ♦t♥ ♥q s♦t♦♥♦r ② ♦♥sr♥ tt ♦♠♠♥t♦♥ ♥♥ ①sts t♥ t t♦ ♥♥♦s ♥ ①tr ♦sr ♦ ♦t♥ ② ♦t ♥♦s rs♦♥ ts ♠t②

30 20 10 0 10 20 3030

20

10

0

10

20

30

anchor node

blind node

t♦ ♦ts r♣rs♥t t ♥ ♥♦s r ♦ts r♣rs♥t t♥♦rs ♥s tr♠♥ t ♥♦rs ♥ ♥ ♥♦s ♦♥♥t♦♥

❯s♥ ♠①♠♠ ♦♦ st♠t♦r t♦ s♦ tt ♥r ♥ ♣r♦♠ ♦ tr t♦ ♥ rr♦r ♥ t ♦rst s ♦r t♦ ♣♦st♦♥ st♠t t♥ t ♦tr♥ ♥♦rs ♥ t st s ❯s♥ t P ♠t♦ ♣r♦♣♦s ♥ st♦♥ tt♦ ♣♦ss s♦t♦♥s r ♣r♦♣♦s r s♦s tt stt♦♥

❲t♦t ①tr ♦♠♠♥t♦♥ t ♠t② r♠♥s ♦r ♦t ♥ ♥♦s s♦t♦♥ ♦ t♦ s t♦ ♠t♦ ♣r♦♣♦s ♥ r t ♥♦r♠t♦♥ s st♦ t♥ ♥ ts ♦♥rr♥t st♠t ♣♦st♦♥s ♦ ♥ ♥♦s

❯s♥ ♦♠♠♥t♦♥ t♦ s♦ t ♠t②

r♦ ♦srs r rt s♥ ♣t ♦ss s♦♥ ♠♦ s sr♥ t t ♦♦♥ ♣r♠trs

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

30 20 10 0 10 20 3030

20

10

0

10

20

30

anchor node

blind node

es timated pos ition blind node 1




rsts ♦ P ♣r♦ss♥ ♦♥ t s♥r♦ P ♣r♦ss♥ s t♦t st♠t♦♥ ♦ ♣♦ss ♣♦st♦♥s ♦r ♥ ♥♦

tr♦♥♦s ♦③t♦♥ ❯s♥ ②♣♦tss st♥

❱r♥ ♦ t t ♥♦s σX = 4dB Pt♦ss ①♣♦♥♥t np = 3 P0 = −40dB

rst t st♥ t♥ t ♥ ♥♦s s ♦♠♣t ♥ t r ♣♦r s r♦♠ t ♠♦ t t ♥ ♣r♠trs

ss♠♥ tt r♥ t ♦♠♠♥t♦♥ ♥ ♥♦ sr ♦t ♣♦r ♥♦r♠t♦♥ ♥ ♦♥ ♦ tr st♠t ♣♦st♦♥s r s♦s t rst♥ ♦③t♦♥st♠t♦♥ st♠t ♣♦st♦♥ s ♦t♥ ② s♥ ♥ tr♦♥♦s ♦③t♦♥ ♦rt♠ s ♦♥ ♠①♠③t♦♥ ♦ t ♦♦ ♥t♦♥ s♥ ♥♠r♦ ♦♣t♠③r ❬❪

40 20 0 20 40 6040

20

0

20

40

anchor node

blind node

poss ible pos ition blind node 1






ML es timated pos ition blind node 1

ML estimated pos ition blind node 2

♦♠♣rs♦♥ ♦ t ♣♦st♦♥ st♠t t t P ♥ t ②♣♦tsstst♥ ♠t♦ t♦ ♥ ts stt♦♥ t ♣r♦♣♦s ♠t♦ r t rr♦r

♦♠♣rs♦♥ ♦ Pr♦r♠♥s

♦♠♣rs♦♥ ♦ ♣r♦r♠♥s ♦ t ♣r♦♣♦s ♣♣r♦ ♥ s ♣r♦r♠tr♦ ♦♥t r♦ s♠t♦♥ t 500 ♥♦s r③t♦♥s s♥ t ♦♠♠♥t♦♥s♥r♦ t t ♣r♠trs sr ♥ t

r ♦♠♣rs t rr♦r t♥ t st♠t ♣♦st♦♥ ♥ t r♦♥ trt♣♦st♦♥ ♦r t ♣r♦♣♦s ♠t♦ s ♦♥ ②♣♦tss tst♥ ♥ s♥ tr t♠♥ ♣♦st♦♥ ♦t♥ r♦♠ t ♦s ♦r ♦♥ ♦ t ♦t

♦r t ♣r♦♣♦s ♠t♦ t ♥ ♦sr tt ♦ t ♥ ♥♦ ♣♦st♦♥sr st♠t t ♥ rr♦r ♦r t♥ ♠trs rs ♦ t st♠t ♣♦st♦♥

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

Pr♠trs stt♥s

Pr♠tr ❱

r♥ r♥ σr 0.5m♣♦r r♥ σX 4dB♣t ♦ss ①♣♦♥♥t np 3♣t ♦ss t ♠tr PL0 4dB

0 5 10 15 20 25 30

Positioning error [m]

0.0

0.2

0.4

0.6

0.8

1.0

CummulativeProbability

Hypothesis testing

ML - chosen position of

the power observable

ML - average position of

the power observable

♦ ♣♦st♦♥♥ rr♦r s♥ t ♣r♦♣♦s ②♣♦tss tst♥ ♣♣r♦ s♥ tr t ♠♥ ♣♦st♦♥ ♦t♥ r♦♠ t ♦s ♥ s♥ ♦♥ ♦ t♦t ♣♦st♦♥ ♦t♥ r♦♠ t ♦s

♦♥s♦♥

r ♦r ♠trs ② ♦♥strt♦♥ t ②♣♦tss tst♥ ♠t♦ ♦s t♥ ♦♥rr♥ts st♠t ♣♦st♦♥s ♦♥ ♦s t♦ t r♦♥ trt ♣♦st♦♥ ♥ ♥ ♠r r♦♠ t r♦♥ trt s ♦♥ s t s♦♥ s ♦rrt t rr♦r ♦♥ t ♣♦st♦♥st♠t♦♥ s ♦♥② t♦ t rr♦r ♦♥ t r♥ ♦srs ❲♥ t s♦♥ tsts t r♦♥ ♣♦st♦♥ st♠t♦♥ s ♦s♥ ♥ t rr♦r s ♥ t ♦rr ♦ ♠♥t♦ t r♥ ♦sr

♥ s ♦ t ♠①♠♠ ♦♦ st♠t♦r t ♣♦st♦♥ st♠t♦♥ s ♣rtr② t♦rs

rst t ♣♦st♦♥ ♦ t ♥♦ s t ♣♦r ♦srt♦♥ s ♥♦t ①t②♥♦♥

r② ♦ t r♥ ♦srs s r ② t ♣♦♦r r② ♦ t♣♦r ♦sr

♦r tt ♣r♣♦s strts ♥ ♥s rst ♦♥ ♦ t t♦ ♣♦sss♦t♦♥s ♥ ② t r♥ ♦srs s s s t ♥tr ♣♦st♦♥ ♦ t s♦♥ ♥♥♦ ts strt② s q♥t t♦ t♦s ♦♣t ♦r t ②♣♦tss tst♥ ♠t♦♦♥ t ♠♥ ♣♦st♦♥ ♦ t t♦ ♣♦ss s♦t♦♥s ♥ ② t r♥ ♦srss s s t ♥tr ♣♦st♦♥ ♦ t s♦♥ ♥ ♥♦ t♦ strts ♥rt♥ts ♥ rs rst strt② rs t ♦ r♠ rr♦r t s♦ r② r rr♦rs ♦♥trr② t s♦♥ strt② ♠ts t♦s r rr♦rs t ♦ ♥♦t t♦ ♣♦st♦♥ t ♥ rr♦r ss t♥ m

❲tr t strt② t ♣r♦♣♦s ②♣♦tss tst♥ ♠t♦ ♣rs ♦♥ t ♠t♦ ♣♦ss ①♣♥t♦♥ ♦ t s♣r♦rt② ♦ t ②♣♦tss tst♥ ♣♣r♦s s ♦♥ t r♥ ♦ r② t♥ r♥s ♥ ♣♦r ♦srs t sstt tt t ♣♦ss ♣♦st♦♥ st♠t♦♥ r② r♦♠ r♥ ♦srss ♥r② r t♥ t ♦♥ r♦♠ ♣♦r ♦srs r♦♠ tt stt♠♥t t ♥ ♥s t♦ s ♦t ♥♦r♠t♦♥ r♥t②

t ♣rs ♣r♦♠ ♥ s♦ s♦ s♥ ♠ss ♣ss♥ t♥qs s ♣r♦♣♦s ♥ ❬❪ ♥ ♥ ♥♦ ①♥s ts ♥t♦♥ t t ♦tr ♦♥r t ♥t♦♥ ♥ ss♠t t♦ ♣♦r ♥st② ♥t♦♥ Prt② t ①♥ ♦ ♥t♦♥ rqrs t♦ s♥ r t♦r ♣♦t♥t②♥rt ♥ ♦r ♥ t ♦♠♠♥t♦♥ ♥♥ ② s♥ t ②♣♦tss tst♥♠t♦ t ①♥ ♥♦r♠t♦♥ s ♠t t♦ ♥ st♠t ♣♦st♦♥ ♥ t ♣♦rr s ♥♦t t♥ ♥t♦ ♦♥t ♥ t ♦♠♠♥t♦♥ s t s ♠sr② t ♥ ♥♦ ♥ ♦ ♥♦t rqr ♥② ♦♠♠♥t♦♥

♦♥s♦♥

s ♣tr s ♥ t ♥ ♦r ♦ ♦③t♦♥ t♥qs ♥ tr ss♦t ♦rt♠s ♥ s♦♥ ♣rt ♥♦ tr♦♥♦s ♦③t♦♥ ♦rt♠ s♦♥ ♦♠tr ♣♣r♦ P s ♥ ♣rs♥t ♥ s♠t♦♥s s♦♥ ttt ♠t♦ ♦t♣r♦r♠ ♣♣r♦s ♥ r♥t ②r s♥r♦s ♥ ♦ t ♥t ♦ t ♦rt♠ s ts t② t♦ rtr♥ ♠t♣ st♠t ♣♦st♦♥s ♥ s ♦ ♦ ♦srs t ♥t s ♥ s ♥ tr ♣rt t♦ ♦t♦♥

♣tr ♥♦♦r ♦③t♦♥ ♦st ♥ tr♦♥♦s ♣♣r♦s

♠t♦ ♦♥ ②♣♦tss tst♥ ♥ s ♦ ♥r tr♠♥ ♣r♦♠s r ♦s♥ t♦ ♠t ♠♦ s♦t♦♥s t ♠t♦ ♣r♦♣♦ss t♦ s t ♦srs t♦ t♥ t♦s r♥t s♦t♦♥s ♠t♦ s ♥ t t ♦♥tr♦ s♠t♦♥s ♥ tst ♦♥ ♥♦♥ ♠ss ♣ss♥ s♥r♦

♣tr

②♥♠ Pt♦r♠ ♥♥Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥♥t ♦t② ♥ t t♦r

♥tr♦t♦♥

♥ ♦rr t♦ ♣r♦ s♠t♦r t♦ rss t ♥♦♦r ②♥♠ ♦③t♦♥ ♣r♦♠s ♥ ♥ rt tr♦♠♥t ♣r♦♣t♦♥ ♥ t ♦③t♦♥♦rt♠s s ♠♥t♦r② t ♥♦t ♦♠♣t② s♥t rst ss♠♥ ②♥♠ s♠t♦♥ rst sr♣t♦♥ ♦ t ♥t ♠♦t② s t♦ t♥ ♥t♦ ♦♥t t♦♥sr ♣r♦♥ t r♦ trs ♦♠♣r t♦ rst stt♦♥s ♦♥ tt ♥rs♥ ♥♠r ♦ r♦ st♥rs ♦♥ s♥ ♠♦ s t ♦ ♥ sst t♦ s♠t ♦ s♦♥ ♦ t s t♦ ♦rt strts strt ♦♠♣tt♦♥ ♦r rt ♥♦r ❬❪ ♥ ♥sr ttr s ♦ t ♥♦r♠t♦♥ ♥ t♦s ♥t♦rs r s♣♣♦s♥ tt t ♣♦st ♥t♦rs ♦ ♥rs qt② ♦ t ♥t♦r ② ♣r♦♠♦t♥ t s②♥r② ♦ t♦♠♠♥t♦♥ ♥ t ♦③t♦♥ t s♠t♦♥ ♦ ♦♠♠♥t♦♥ ♥♥ ♦♥sr t♦ ♦♠♣♥t t t ♣♦♠♥ ♦t♦♥s s ♣tr ♠s t♦ ♦♠♣t r♠♦r ♦♥ t s♠t♦♥ ♦ t ♥♦♦r ②♥♠ ♦③t♦♥ ♣r♦♠s ♦r tt ♣r♣♦s ♥ t♦♥ t♦ t ♠t♦s ♣rs♥t ♥ t ♣r♦s ♣trs ♠♣♦rt♥t s♣ts ♦ t ②♥♠ ♣t♦r♠ r ♦♣

t ②♥♠ ♠♦t② ♦ t ♥t t r♦ ♥t♦r ♦ ♠♦♥t♦r♥ t ♥tr ♥ts ♦♠♠♥t♦♥ rst st♦♥ ♣r♦s stt ♦ t rt ♦ ♦t ♠♦st s ♠♦t② ♠♦♥

♥ ♦ ts ♠♦t② s t♥ ♥t♦ ♦♥t ♥ s♠t♦rs s ♣r♦ s♦♥st♦♥ ♣rs♥ts t ♦s♥ t ♠♣♠♥tt♦♥ ♦ t ♠♦t② ♦ t ♥ts♥t♦ t ②♥♠ ♣t♦r♠ ♥t ♠♦t② s s ♦♥ srt ♥t ♠t♦r ♦♥ ♥t t♦ ♠♦ ♥♣♥♥t② r♦♠ ♦tr ♠♦t② ssr ♦t t r s s♥ r♣ sr♣t♦♥ ♥ t s♠ s s♥

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

t str♥ ♦r ♠t♦ tr st♦♥ srs ♦ t tr♦♠♥ts♠t♦♥ ♥ t Ps r ♠♦♥t♦r ♥t♦ t ②♥♠ ♣t♦r♠ ♥ ♣rtr tr♣rs♥tt♦♥ ♦ t r♦ ♥t♦r s♥ r♣s s t ♥② t♥ ♥t♦ t r♣ ♦r♥③t♦♥ ♦ ♦t t ♥t♦r sr♣t♦♥ ♥ t s♠t♦♥ tst st♦♥ srs t ♥tr ♥ts ♦♠♠♥t♦♥ ♣ts ♦ t ②♥♠♣t♦r♠ st st♦♥ ♣rs♥ts r♥t s♠t♦♥ s♥r♦s t♦ strt t②♥♠ ♣t♦r♠ ♣♦t♥t rst ②♥♠ s♠t♦♥ ♥♦♥ s♥ ♥t s r♥r Ps ♥ r ♦t♥ ♦r r♥t ♣♦st♦♥s s♥ ♦t t ♠t♠♦ ♥ t t♦♦ t s♠t♦♥ ♥♦♥ t♦ ♥♦s s ♣rs♥t t♦♠♦♥strt t ♥tr ♥ts ♦♠♠♥t♦♥ tr

tt ♦ t rt ♦♥ ♦t② ♥ t♦r ♠t♦rs

qst♦♥ ♦ t rst s♠t♦♥ ♦ t ♠♦t② s ♣r♦♠ s ♥①♣♦r ♥ ♠♥② r♥t s s♥ ♠♥② r♥t ♣♣r♦s ♦r t ♣r♦♠ t♥ t ♣♣rs tt ♥ rss ♣r♦♣t♦♥ s♠t♦r t♦ t ♥t♦ ♦♥tt ♠♦t② s♦ ♦s ♦♥ t ♦ ♦ t ♦rrt ♠♦t② ♠♦ ❬❪ ♦rtt ♣r♣♦s ts st♦♥ strts t ♥ ♦r ♦ t ①st♥ s♦t♦♥s ♦r ♠♦♥t ♠♦t② ♥ ♥ t ♥ts ♦ srt ♥t ♠t♦r ♥ t♦♥sr ♦♥t①t r t ♥ ♥② s♦♠ ♣r①st♥ s♦t♦♥s ♦♠♥♥♠♦t② ♥ ♥t♦r s♠t♦♥ r ①♣♦s ♥

②♣s ♦ ♦t② ♠♦s

♦r♥ t♦ ❬❪ ♠♦t② ♠♦s ♥ t♦r③ ♥ tr t②♣s ♥♦♠ ♠♦s ♦s t ♦r♣ rstrt♦♥ ♦s t s♣t♦t♠♣♦r ♣♥♥②♥♦♠ ♠♦s r ② s ♥ ♦♥tr♦ s♠t♦♥s r t ♣r♣♦s

s t♦ ♥tr♦ rst tt♦♥ ♦ t ♥♥ ♦♥ t s♥ ♥ tst ♦♦♠♠♥t♦♥s ♦rt♠s ♦r P❨ s♥ t♦s s♦t♦♥s r r② s② t♦♠♣♠♥t t rst♥ ♠♦t♦♥ s ♥r② ♥♦t r② rst t rs♣t t♦ ♥s♣t r♦♥ trt ♠♦♥ t♦s r♥♦♠ ♠♦s ♦♥ ♥ t♦s s ♦♥ r♦♥♥♠♦t♦♥ ❬❪ ♦r r♥♦♠ ② ♣♦♥t ❬❪ ♦♠ ♦ t♦s ♠♦s s♦ t♠♣♦r♣♥♥② ♥ ♠ ♣♥ t t ♦t② ♦♥ t ♣r♦s ♦♠♣t ♦ts♥ ♦ t ♠♦st s ♠♦ t♦ ts rt sttst ♦r♥ t r ♠♥♠♦t② s t ② t r♥♦♠ ♠♦ ❬❪ ♥ t é② t ♠♦ t st♣sr ♥ ♥ tr♠s ♦ t st♣♥ts rt♥ ♣r♦t② strt♦♥t rt♦♥s ♦ t st♣s ♥ s♦tr♦♣ ♥ r♥♦♠ ②♣② t st♣s ss♠ s♣ Prt♦ strt♦♥ ❬❪ ♦s r♥♦♠ ♠♦s r r② ♦♥♥♥t t♦ t ♥t♦♦♥srt♦♥ t ♦ ♦r ♦ ♥ ♥t ♠♦♠♥t t ② ♥tr t② r ♥♦tst ♦r rss♥ sts♣ stt♦♥s ♥ s♦♠ ♦rs tr t♦ ♥tr♦t ♥r♦♥♠♥t ❬❪ ♣♣r♦s t♦ ♦♥sr t♦ t t tr♠♥sts♠t♦♥ ♦ t ♥♥ ♣r♦♣t♦♥

tt ♦ t rt ♦♥ ♦t② ♥ t♦r ♠t♦rs

♦r s s♠t♦♥s t s ♠♥t♦r② tt t s♠t ♥t ♣♦st♦♥s ♦♥♥ s♣ t s t ♠ ♦ ♠♦s t ♦r♣ rstrt♦♥ ♠ts t ♠♦♠♥t ♦ t ♥ts ♥t♦ rstrt s♠t♦♥ ♥ ♥♦♦r stt♦♥ t ♠♥t tt ♥ ♦♦r ♣♥ ♦ ♦♥ t ♣♦st♦♥ ♦ t ♥ts ♥ ♠t tr ♠♦♠♥tt♦ ♣♦ss ② ♣ts r♦ss♥ t ♦♦rs r♣ s ♠♦s ♦♥ t♦ ts t♦r②② ♦♥str♥♥ t ♠♦♠♥t ♦ ♥ ♥t ♦♥ t r♣ ♥♦s ❬❪ s t ♥tt♥ ♦t② ♠♦ ❬❪ ♠ts t ♠♦♠♥t ♦ t ♥♦s t♦ r♦s t♥♥s ♦s ♦r♣② rstrt ♠♦s r r② ♦♥♥♥t t♦ sr t♠♦♠♥t ♦ ♥ts t t② t♦ t ♥t♦ ♦♥t ♥tr♥t ♥trt♦♥s rqr s②♥r♦♥③t♦♥ ♦r ♠♥s♠

♦r tt ♣r♣♦s s♣t♦t♠♣♦r ♣♥♥t ♠♦s r ♥tr♦ ♦s ♠♦st r ♦ t ♠tr ♥r♦♥♠♥t ♥ t ♠♥ ♥r♦♥♠♥t s ♦rs ♥tr♥t ♦♥ ♦r r♦♣ ♠♦t② t♦ ♦♠♠♦♥ trt ♥ ♥s rst②♥rs♥ t rs♠ ♦ t s♠t♦♥ ❬❪ t ♦ rs♠ s t② t ♦st rqr t♦ sts②♥ s♠t♦♥ ♦ t ♠♦t② ♥ r♦ ♦♥t①t ♥♦♥sr♥ s♠t♦♥ ♥ ss ♦ ♦♦♣rt ♥t♦rs t ♥♠r ♦ ♦♦♣rt♥ts ♥ ♥ r s rt② ♣r♠tr ♦♥ t ♣♦st♦♥ r② ♦ ♥ ♦rt♠ ❬❪

♥♦tr ♠♣♦rt♥t s♣t ♦r ♣r♦♥ ♦rrt s♠t♦♥ s t ts ♦ t♥ts ♥ ♠♥ ♠♦t♦♥ r ② ts ❬❪ ♥ rt② tr♥st♥t♦ t ♠♦t② ♠♦ ♦ t ♥ts ts ♥ ♠♦ r♦♠ t ♣r♦♣♥st②♦ ♥ ♥t t♦ ♥ s♣ r♦♦♠ ♦ ♥ ♦r t♦ ♥r t ♥♥ ♦ ♥♦tr♥t

♥② sts②♥ ♥t ♠♦t② sr♣t♦♥ ♦ sts♣ ♥ ♥s♠rt ♥trt♦♥s ♦ t ♥ts t ♦t t tr ♥r♦♥♠♥t ♥ t t ♦tr♥ts ♦♥♥♥t s♦t♦♥ t♦ s tr♠♥st ♠♦t② sr♣t♦♥ s t♦s srt ♥t ♠t♦r

srt ♥t ♠t♦r

r♦♠ t ♣r♦s st♦♥ t s ♥ s♥ tt t ♠♦t② s ♥tr♥s② s♣t♠ ♣r♦ss ♥ ♥trt♦♥ t t ♥r♦♥♠♥t Prt② t s♠t♦♥ ♦ s ♣♥♦♠♥ ♥ qt ♦♠♣t t♦ rss s t ♥♦s r♥♦♠ rs♣♦ss ♣♦st♦♥s ♥t♦ t ②♦t tr♥st♦♥s ♣r♦ts t♦ ②♥♠② ♠♦ ② ①tr♥ ♥ts ♦tr ♥ts rr ♦s ♣rtr ♥ ♦♣r♦♠s ♥ ♥t♦s② ♠♥ ② s r s♥ ♦r ♦t st♦st ♥ ②♥♠ ♣r♦♠s ♦♥ t srt ♦t♦♥ ♦ rs ❬❪ ♦♦ s r② ♥ ♠ ② ♦tr ♠♦t② s♠t♦rs ♦r♥t ♦♥ ❯r♥ ♠♦t②❬ ❪ ♦r t ♣r♦♠ t ♥ ss♠♥ P②t♦♥ ♠♣♠♥tt♦♥ ♥ts ♠♥sr♣t t ♠P② r♠♦r s ♥ s ❬❪ ♠P② s ♦♥t s♠t♦♥ ♦ t ♦♠♣♦♥♥ts

♥♦tr ♥trst♥ s♣t ♦ s♥ s t ♣♦sst② t♦ ♥ t♦♥♦♠♦s♥tts Prt② t ♠♥s tt t s♦♥s ♦ ♥ ♥t s s ♦♥② ♦♥ ts ♦♥♦srt♦♥ ♦ t ♥r♦♥♠♥t t t♦♥♦♠② ♥ ♠t ② rs ♥ ② ♥

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

sr ♦♥ rt♥ r♦♦♠s ♦ t ♥r♦♥♠♥t t ♣rs ♠♣♠♥tt♦♥ s♥ ♦r♥② rt ♦r ♦ ♠s ♥ t s♠t♦♥ ♦ rt ♥t♥ ♥♠♣♠♥tt♦♥ ♦ t♦s t②♣ ♦ s♦t♦♥ s s ♦♥ t str♥ ♦rs ♣r♥♣s❬❪ tr♥ ♦rs ♦♥ t ♦♠♣tr sr♣t♦♥ ♦ ♠♥ ♦r ♦♦ r tt ♥ ♦ ♦r ♣t ♦♦♥ s ♦♥ ts ♣r♥♣s rr② Prs♦♥ ♣ r♥st ♠t♦♥ ❬❪ s ♥ ♥♥ t♦ t t t♥♦♦r ♦③t♦♥ ♣r♦♠ts ♠♦r ♣rs sr♣t♦♥ ♣r♦ ♥ st♦♥

t♦r ♠t♦♥ ♥ ♦t②

t♦ t ①sts ♦♠♠r t♦♦s s♣③ ♥ t ♣str♥ ♠♦t② t♦ ♠♣r♦♣ss♥r ♦ ♥② ♥ ①st♥ ♥s ♦r t♦♥ ♥ tr ♠♥♠♥t❬ ❪ tr s ♥♦ s t t♦♦s ♥ t rss ♦♠♠♥t♦♥ ♦r♦r ♥ ♦♠♠r ♣r♦♣t♦♥ t♦♦s ♠♥t♦♥ ♥ st♦♥ t ♠♦t② s♥r② ♥♦t t♥ ♥t♦ ♦♥t t ♦ s♦ ♠♥t♦♥ ♦♥sr♥ ♣♣r②rs ♦ t st♥rs P r t ♠♦t② ♣r♠trs s rr② ♣rt♦ s♠t♦rs ❬ ❪ t♦ ♠♦st ♦ ♦♠♠r t♦♦s t♥ ♥t♦ ♦♥t t♠♦t② t ①sts s♦♠ rsr t♦♦s ♥trt t ♥t♦ t ♥t♦r ♣r♦r♠♥st♦♥ ♦r tt ♣r♣♦s ♥ ♦♥sr♥ ♥♦♦r stt♦♥ t♦ ♠♦s r ♥r②s

r t♦♠t ♠♦s r s ♦♥ ♠♦ t ♦r♣ ♦♥str♥ts ♥t s ♠♦s r tr s♣t♦t♠♣♦r ♣♥♥② ♠♦s ♦r s ♠♦

r ♥♠r ♦ t♦s sts ♦t t ♥♥ ♦ t ♠♦t② ♦♥ t ♥t♦rs r t♦♠t ♠♦s ❬ ❪ s ♠t♦ ♦♥ssts ♥ t srt③t♦♥ t♥t ♣♦st♦♥ ♦♥ r t t♠ st♣ ♥t ♥ ♠♦ t♦ ♥t ♥♠r♦ ♣♦ss ♣♦st♦♥s ♦♥ tt r t♦ t s ♦ t ♠t♦ t s ♦ r ♦r♠♦♥ t ♠♦♠♥t ♦ t ♥t s ♦t♥ ♦♥sr s ♠♦t② ♠♦ ❬❪ ♥♦rr t♦ ♠♦r sts♣ s♦♠ ♣♦st♦♥s ♦♥ t r ♥ ♦r♥ ♣r♥t♥♥ts t♦ st♥ ♥ ♥ ♦♣♥ ♦♦r ♦r ♥st♥ ❬❪

♠♦r s♦♣stt ♣♣r♦ s t ♥t s ♠♦ ♥t♥s t♦ r♣r♦ r ♣str♥ tt② ♦♥str♥t ② t ♥r♦♥♠♥t ♥ tt s t s ♦ s ♠♦t② ♠♦ ♦s t♦ rtr ♥ s♠t♦♥ ♣♥♦♠♥ ♥♥♦t tt t s♠♣ r♥♦♠ ♦r r t♦♠t ♠♦ ❬❪ s♣t ♦tr ♣♦t♥tts ♦♥② s♠t♦rs s tt ♠t♦ ♥ t ♦♥sr ♦♥t①t ♦r♦ s♠t♦♥s ❬❪ ♦r♦r t qst♦♥ ♦ t ♦♠♠♥t♦♥ ♦r strt♦♠♣tt♦♥ ♥♥ t ♠♦t② s♠s t♦ ♥

♠t♦♥ ♦ t ♥ts ♦t②

♣r♦♣♦s s♠t♦♥ ♦ t ♥t ♠♦t② s s ♦♥ ♠♦♠♥t♦ t ♥ts sr♣t♦♥ s ♥t♦ t♦ ♠♦s rst t r s ♠♦t② r♣rs♥ts t ♠♦tt♦♥ ♦ t ♥t ♦♦s♥ st♥t♦♥ r♦♦♠ s sr

♠t♦♥ ♦ t ♥ts ♦t②

♥ ♦♥ t s♠ s ♠♦t② ♦♥str♥ts t ♥t t♦ ♠♦ ♥♥trt t t ♥r♦♥♠♥t ♥ t ♦tr ♥ts s sr ♥

r ♦t② r♣ sr♣t♦♥

t r s t ♥ts ♠♦t② ♥ s♥ s st♦st sss♦♥ ♦ t♦r♥t stts

stt r t ♥t s stt stt r t ♥t s ♠♦♦st ♦ t t♠ ♥ ♥♦♦r ♥r♦♥♠♥t ♠♦ ♥ts r stt ts t stt

stt ♥ ② ♦♥sr s ♣rt ♦ t ♦r ♠♦t② sr♣t♦♥ r s♠♦t② ♦♥ssts ♥ ♦t ♥♥ trt ♦r t ♥t ♥ tr♠♥♥ ♣t t♦ rtt trt

r♣ ♦ ♦♦♠ Gr ♥ t r♣ ♦ ❲② Pt Gw

♥ ♦rr t♦ tr♠♥ t r♥t trts ♦ t ♥ts t r♣ ♦ r♦♦♠ Gr s♥tr♦

0 1 2

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4

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♥ ①♠♣ ♦ r♣ ♦ r♦♦♠s Gr

r♣ ♦ r♦♦♠s Gr r♣rs♥t ♥ r s s♠♣ rs♦♥ t♦♣♦♦r♣ Gt sr ♥ s♣② s♥ ♦r t ♠♦t② ♣r♣♦s ♥♦s ♦t Gr r♣rs♥t r♦♦♠s ♦r ts ♣r♣♦s t ♥♦s ♦ Gr r t ♥♦s ♦ Gt ♠tt♦ ②s Gts ♥♦s t ♦♦r ♥ t♦♥ ♦ Gr ♥tr♦♥♥ts r♦♦♠ssr♥ ♦♦r ❲t s sr♣t♦♥ ♥♦ ♥ tr♠♥ trt ② ♦♦s♥ r♦♦♠ ♥ t ♥♦s ♦ Gr ♦t t trt ♥ t ♣t r tr♠♥ t t ♣♦ r♣s ♦r ♠♦r rst ♠♦♠♥t t r♣ ♦ ② ♣t Gw r♣rs♥t♥ t ♣♦ss tr♥st♦♥s t♥ r♦♦♠s s ♥tr♦

0 1 2

3

4

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101112

13

14 1516

31

72 77

78

80

83

85

87

88

90

93

94

96

99

103

101

♥ ①♠♣ ♦ r♣ ♦ ② ♣t Gw ss♦t t♦ t r♣ ♦ r♦♦♠

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

r♣ Gw r♣rs♥t ♥ r s t s♠ r♦♦♠ ♥♦s t♥ Gr ♣s①tr ♥♦s ss♦t t♦ ♦♦rs ♦s ♥♦s r t♦ ♥♦r t ♥t♦t t♦ r♦ss t ♦♦rs ♥ t♦ ♦ ♥r trt♦r② ♦♥ t♦ t ♥tr ♦ ♦rr♦r ♥st ♦ rt② r♥ t ♦♣♣♦st r♦♦♠ ♦s r ♦♥♥t s♥ t♦rs

r♦♦♠ ♥♦s r ♦♥♥t t♦ ♦♦r ♥♦ ♦♦rs ♦ t♦ r♦♦♠s r ♦♥♥t t♦tr t r♦♦♠s r ♦♥♥t ♥

Gr

tr♠♥t♦♥ ♦ rt ❯s♥ t r♣s

Prt② t r♣ Gr ♦s t ♥t t♦ r♥♦♠② ♦♦s trt st♥t♦♥ r♦♦♠ ♥ t r♣ Gw ♦s t♦ ♥ ♣t t♦ ts trt t ♣t s sss♦♥ ♦ Gw ♥♦s

ts s♣♣♦s stt♦♥ r ♥ ♥t ♥ r♦♦♠ i ♦♦ss trt r♦♦♠ t tt ♣ ♦ r♣ Gr ♦♦♠s i ♥ t ♦rrs♣♦♥ rs♣t② t♦ ♥♦s vir ♥ v

tr ♦ Gr

♥ t ♣t r♦♠ t r♦♦♠ i t♦ t r♦♦♠ t s ♦t♥ ② ♣r♦ss♥ str ❬❪s♦rtst ♣t ♦rt♠ D ♦♥ ♥♦s viw♥ v

tw ♦ t r♣ Gw

Vp = D(viw, vtw).

♦t♥ ♦rr st ♦ ♥♦s Vp srs t s♦rtst ♣t t♥ viw ♥vtw ♦ t ♦ trt♦r② r♦♠ i t♦ t t ♥t s t♦ sss② ♦ tr♦ t r♦♦♠s ♥♦r ♦♦rs ♦rrs♣♦♥♥ t♦ t ♥♦s ♦ Vp t♠ ♥♦ ♦ Vp

s r t ♥①t ♦♥ ♦♠s ♥ ♥tr♠t trt it ♥t t ♥♦ vtr s r♥ r ♥ ♥t ♥ r♦♦♠ i = 12 t trt ♥ r♦♦♠ t = 15 ♦rrs♣♦♥s

t♦ rtr♥ ♣t Vp = 12, 31, 93, 15 s t ♥t rst ♠♦s r♦♠ r♦♦♠ 12 t♦ ts♥tr♠t trt 31 ♥ s♦ ♦♥ ♥t t ♠♦s t♦ ts ♥ trt ♥ r♦♦♠ 7

♠ ♦t② tr♥ ♦rs

♥ tr trts r ♥♦♥ ♥ tr ♣ts ♥ tr♠♥ ♥ts ♥ strt♠♦♥ ♥ t ♥r♦♥♠♥t ♦♥ s ♥ ♦♥ tr♦ ♦♦rs s ♣rtrt② s ♥ s t s♠ s sr♣t♦♥ ♦ t ♠♦t② ♥ s sr t t♣ ♦ str♥ ♦rs

tr♥ ♦rs ♥ ♥tr♦ ♥ ❬❪ t♦ sr ♥trt♦♥s ♦ ♥tst tr ♥r♦♥♠♥t ♥ts r s♠♣② sr ② tr ♠ss m ♥ t♦ ♠t♥♣r♠trs tr ♠①♠♠ rt♦♥ amax ♥ tr ♠①♠♠ ♦t② vmax ssr♣t♦♥ ♦ t ♥ts t♦ r♥ ② sr str♥ ♦rs ♣♣ ♦♥ tr♥tr ♦ ♠ss r♥t str♥ ♦rs r sr ♥ st♦♥s t♦

♦♠♣tt♦♥ ♦ rt♦♥ ♦t② ♥ ♣♦st♦♥ ♦ ♥ts s ♥sr ② r♥trt♦♥ ❬❪ t t♠ st♣ t ♥ts rt♦♥ a s rt ♥ rrst t rst♥ str♥ ♦r ♣♣ ♦♥ t tr str♥ ♦rs s tr♥t♥ rr ♦ t ♠①♠♠ rt♦♥ ♣r♠tr ♦ t ♥t amax str♥ ♦rF t② ♣♣ ♦♥ t ♥t s ♥ ②

♠t♦♥ ♦ t ♥ts ♦t②

F := max (||F||,mamax)F

r F s t ♥tr② t♦r ♦t♥ r♦♠ ♥♦r♠③t♦♥ ♦ F ♥ m s t ♠ss ♦t ♥t ♥ t ♣♣ rt♦♥ t♦r an t t♠ st♣ n s ♥ ②

an =F

m

s t tr ♦t② vn t t♠ st♣ n s ♣♣r♦①♠t ② t r ♥trt♦♥s t s♠ ♦ t ♦ ♦t② ♥ t ♣r♦t ♦ t rr♥t rt♦♥ t♦r tδτ t t♠ t♥ n− 1 ♥ n

vn = vn−1 + anδτ

♥ t ♣♣ ♦t② vn s tr♥t ② t ♠①♠♠ ♦t② ♣r♠tr ♦t ♥t vmax

vn := max (||vn||, vmax)vn

r vn s t ♥♦r♠③t♦♥ ♦ vn ♣♦st♦♥ pn t t♠ st♣ n s ♦t♥ ② t r ♥trt♦♥ s t s♠ ♦

t ♣r♦s ♣♦st♦♥ pn−1 ♥ t rr♥t ♣♣ ♦t②

pn = pn−1 + vnδτ

str♥ ♦rs ♣♣ ♦♥ t ♥t ♦rrs♣♦♥ t♦ ♠♥ ♦rs ♥t ♦♦♥ ♦♥② tr t②♣s ♦ str♥ ♦rs r ♦♥sr t s ♦r t♦st ♦♥ ♦r ♥ t ♥t ♦♥ ♦r

♦r

s ♦r ♠s t♦ ♣r♦ str♥ ♦r ttrt t ♥t t♦ trt ♥ t ♦♥sr s♥r♦ t♦s trts r sss♦♥ ♦ ♥tr♠t trtss ♥ ♥ st♦♥ str♥ ♦r rt♦♥ ♣♥s ♦♥ t st♥ t♦rdn t♥ t trt tn ♥ t t ♣♦st♦♥ ♦ t ♥t pn ♥ t s ♣♦sst♦ tr♠♥ ♦t② t♦r vdn ♥ t

dn = tn − pn

vdn = max

( ||dn||δτ

, vmax

)

dn

r vn ♥ vdn r t rr♥t ♦t② t♦r ♥ t sr ♦t② t♦r rtrs♣t② ♥ dn s t ♥♦r♠③t♦♥ ♦ dn

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

targetagent

♦r str♥ Fs t♦r rsts r♦♠ t rr♥t ♦t② t♦r♦ t ♥t vn ♥ t sr ♦t② t♦r vdn rt t♦ t trt

s t ♦t② r♥ wn t♥ t t ♦t② vn ♥ t sr♦t② vdn ♦s t♦ ♦♠♣t t s♥ ♦r Fs t♦ ♣♣ ♦♥ t ♥t

wn = vdn − vn

Fs = mwn

δτwn

r wn s t ♥♦r♠③t♦♥ ♦ wn ♥ m t ♠ss ♦ t ♥t s stt♦♥s r♣rs♥t ♥ r

st ♦♥ ♦r

♦st ♦♥ ♦r ♠s t♦ ♣r♦ str♥ ♦r ♦♥ t ♥t♣♥trt♥ ♦r ♥② ♣rt ♦ t ②♦t ss♦t str♥ ♦r Fo s ♥②

Fo(dk,w) =m

δτ

(

(vn−1 − vn)α

|dk,w|n

)

,

t dk,w t st♥ t♦r r♦♠ t ♥t k t♦ w α ♣r♠tr ♣r♦♣r②♦s♥ ♦r ♦♥ t ♥t r♦♠ r③♥ t ♥ n ♥♦r♠③ t♦r ♦rt♦♦♥t♦ t ♦st

♥t ♦♥

♥ ❬❪ t s stt tt t ①sts ♥ ♥tr♣rs♦♥s st♥ ♦♥r② s ♦♥r②♥ ♠♦ t r ♦ rs r r♦♥ t ♥t k ❬❪❬❪ ❲ ♥♦tr♥t l ♣♥trts ♥s tt r ♥ ♦♥ ♦r Fa ♣r♦♣♦rt♦♥ t♦ t st♥t♦r t♥ t ♥ts dk,l s ♣♣ t ♦r ♥ ♠♦ s♥ t ♣r♦s②♥ ♦st ♦♥ ♦r rr♥ t ♠♦t♦♥s

st♥ t♥ t ♥t ♥ t ♦st s r♣ ② t st♥st♥ t t♦ ♥ts

r♦tt♦♥ ♠tr① s t♦ ♥tr♦ ♥ ♦rr tt t ♦r ♣♣ ♦rt♦♦♥② t♦ t ♦t② t♦r ♥ ts rt♥ ♥ ♦♥ ♦ ♦t ♥ts

s t ♥t ♦♥ ♦r Fa ♥ rtt♥

Fa(dk,l) =

HFo(dk,l) , ||dk,l|| ≤ r~0 , ||dk,l|| > r

♠t♦♥ ♦ t ♦ t♦r r♥③t♦♥

t H r♦tt♦♥ ♠tr①

H =

(

0 −11 0

)

st♥ ♦r

♥ ♥t k s ♥r t ♥♥ ♦ r♥t str♥ ♦rs s ♦r ♣ t ♥t t♦ ts ♥tr♠t trt ♥♦ vit s s♥ ♥ ♥ sr r♣s♦rs t♦ ♦ ♦sts ♦r ♦tr ♥ts rst♥ str♥ ♦r F s rtt♥ st s♠ ♦ t str♥ ♦rs ♣♣ ♦♥ ♥t k

F = Fs +

W∑

w

Fo(dk,w) +K∑

l 6=k

Fa(dk,l),

r W ♥ K r rs♣t② t ♥♠r ♦ s ♥ ♥ts ♥ t ♥t② ♦ t♥t k s tr♠♥ t s♠♣ st♥ trs♦ s ♦ tt trs♦♦s t sss ♦♠♣tt♦♥ ♦ ♦rs r♦♠ t s ♦ t ♥tr ②♦t

♠t♦♥ ♦ t ♦ t♦r r♥③t♦♥

♥♦tr rqr♠♥t ♦r ♥ ♥ ♥t ②♥♠ ♣t♦r♠ ♦r ♦③t♦♥ st♦ s♠t t ♥t♦r ♦r♥③t♦♥ ♦ r♦ s ♠♦♥ ♦tr t rqrs t♦

t♦ tr♠♥ t r♦ ♦srs s t rs♣t t♦ t ♥t ♣♦st♦♥ t♦ ss♥ t ♦rrt ♦srs s t♦ t ♦rrs♣♦♥♥ r♦ ♥s t♦ s♠t t rrs t♠ ♦ t♦s ♦srs

♠r② t♦ t ♠♦t② t ♦ ♦ s ♥ ♠ t♦ t♦ ♣t t ♣r♠tr s ♦ ♥② ♥t tr ♥♣♥♥t② ♦r t s♣ rrs t♠♥ ♥ ♥ ♦♦s ♦ t♦ sr ts ♥t♦r ♦r♥③t♦♥ s t♦ t ♥t♦ r♣s ♦r tt ♣r♣♦s t ♥t♦r r♣ Gn s ♥tr♦ r♣ ♥♦tt♦♥s ♥ tt ♣rt s ♦♠♣♥t t t ♥♦tt♦♥ ♥tr♦ ♥ sst♦♥ ♥①♠♣ ♦ ♥t♦r r♣ s ♥ ♥ r

t♦r r♣ r♥③t♦♥

♥t♦r r♣ Gn s trtr♣ tr♣ ♠♥s tt t♦♥♦s ♥ ♦♥♥t t♦tr t ♠♦r t♥ s♥ s ♦ ♠t♣s ♦s t♦ r♣rs♥t t ♣♦ss ♠t♣ s ♥♥ t r♦ s s ♦ t s♠t♦♥ r R = ρ1, . . . , ρNr−1 r ρr s ♥ s t r♣ s t s ♥ ♦r♥t r♣ ♠♥s tt t ♥♦r♠t♦♥ ♦♥t ♥♥ t♦ ♥♦s A ♥ B s ♥♦t ♥ssr② t s♠ s tt ♦♥ t ♥♥ B t♦ A ♥ t tr♥s♠ss♦♥ ♥♥ s r♣r♦ t s rqr t♦ t♥t♦ ♦♥srt♦♥ t r♥t ♠ss♦♥ ♣♦r ♥ s♥stt② ♦ ♥♦s

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

①♠♣ ♦ ♥t♦r r♣ Gn t ♥♦s ♦♥ r♥t s ②♦tt ♦s s♣② ①tr ♥♦r♠t♦♥ ss♦t t♦ t r♣ s ♥ ♥♦s

♦s ♦ Gn

♥♦s ♦ Gn rtt♥ Vn r s♥t ② t ♦ tr rrr♥ ♥t ②♦♥t♥ ♥♦r♠t♦♥ ♦t tr r♦ ♣ts ss♠♥ t r♣ tstrtr sr ♥ ♥♦ s t ttrts

②♣ t②♣ ttrt ♥ts t ♥♦ s ♥ ♥t ♦r s stt♦♥ ② ♦♥♥t♦♥ s stt♦♥ s r♦ ss ♣♦♥t s stt rs ♥ ♥t s r♦ ss ♣♦♥t ♥ ♠♦

s s ttrt Rk s sst ♦ R tr t ♦ t k Rk ⊂ R

♠tt ♣♦rs ttrt srs t ♠tt ♣♦r ♥ ♥ ♥t② r♥t ♦r ♦r ts ♣r♣♦s t ♠tt ♣♦r ttrt s t♦♥r② rt ②s r s st ♠♦♥ ♦♥ ♦ t♦s ♦♥ t ♥ t t ♠tt ♣♦r ♦ t ss♦t ♥ dBm

♥stt②♠r② t♦ ♠tt ♣♦r s♥stt② ♥ r♥t ♦r ♥t ♥ ts♥stt② ttrt s s♦ sr t t♦♥r② r t ②s r sst r♦♠ t♦s ♦♥ t ♥ t t s♥stt② ♦ tss♦t ♥ dBm

r♦♥ trt ♣♦st♦♥ r♦♥ trt ♣♦st♦♥ s ♦t♥ r♦♠ t ♠♦t② ♣rt sr ♥

st♠t ♣♦st♦♥ st♠t ♣♦st♦♥ s ♦t♥ s♥ ♦③t♦♥ ♦rt♠s sr ♥ st♦♥

Prs♦♥♥ ♥t♦r

♠t♦♥ ♦ t ♦ t♦r r♥③t♦♥

♣rs♦♥ ♥t♦r s r♣ t r♦♠ t ♥t s♦♥ ♦ t ♥t♦r ♦rts r ♥ ♥ st♦♥

♠ st♠♣ t♠ st♠♣ ♥ts ♥ t ♣r♠tr s ♥ rrsrt② t♦ t ♥♥♥ ♦ t s♠t♦♥

s ♦ Gn

♥ t s En ♦ Gn r st♦r ♥♦r♠t♦♥ ♦t ♥♦s ♦♥♥tt② ♥ s tr ttrts r

P t♦♥r② r♦♥ trt st♥ ❱st②s ♠♥t♦♥ ♦r Gn s ♠t♣ r♣ r ♣r r♣rs♥ts

s♥ ②s ♦ t P t♦♥r② r t Ps ② t ♥t s ♦ t♦s Ps r ♦♠♣t t t ♣ ♦ t ♠t ♠♦ sr ♥ ♥ t tr tt t♦♥ ♦ ♥t♦s② ♣r♦r♠ ② t t♦♦ st② ♣r♠tr s ♦♦♥ tr♠♥ ② ♦♠♣r♥ t s♥stt② t♦ t ♦♠♣t P

t♦rs

♦r ♦♠♣tt♦♥ ♥② t s ♦♥♥♥t t♦ tr ♥♦s sr♥ t s♠ ♥t♦ s♣ ♥t♦r s♦ s♥t♦r s♥t♦rs r t♥ t r♦♠t ♦ Gn ♠tr♣ ② rt♥ s ♠♥② r♣s s Gn s Prt② s♥t♦r s ♥ s t♦♥r② t ρ s ② ♥ t sr♣ ♦ Gn(ρ) s sr♣ Gn(ρ) s Gn rstrt t♦ ts ♥♦s vkn(R

k) ♥ t ρ s ♥ Rk

♥ ♥ rtt♥

Gn(ρ) = Gn −K⋃

kρ 6⊂Rk

vkn(Rk )

Prs♦♥ t♦r Gp

♥ ♦rr t♦ t♦ rss strt ♦r ♦♦♣rt ♦③t♦♥ s♠s ♥♦s♦ ♥t♦r s♦ tr ♦♥ s♦♥ ♦ t ♥t♦r ♦r tt rs♦♥ t ♦♥♣t♦ ♣rs♦♥ ♥t♦rs s ♥tr♦ ♣rs♦♥ ♥t♦r s t r♣rs♥tt♦♥ ♦ Gnr♦♠ ♥♦ ♣♦♥t ♦♥trr② t♦ t ♥t♦r r♣ ♦♥ ♥ st ♥♦♠♣t ss t♦ t ♣♦st♦♥ ♥ Ps ♦ t ♥♦s ♦ t ♥t♦r ♣rs♦♥♥t♦r s rst ♦ t t ♥♦r♠t♦♥ ♣♦ssss ② ♥♦s ♦♥ t♦s ♥♦r♠t♦♥ ♥♦ ♥ ♣r♦r♠ ts ♣♦st♦♥ st♠t♦♥ ♥q②t ts ♦♥ ♥♦r♠t♦♥ ♣rs♦♥ ♥t♦r ♦ ♥♦ k s s♥t ②Gkp Prt② t ♣rs♦♥ ♥t♦rs Gk

p r t s ♠trtr♣ r s

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

s ♠ ♣rs♦♥ ♥t♦r t♥ Gn s ♥♦s ♥ ①♠♣ ♦ t ♣rs♦♥ ♥t♦rsss♦t t♦ r♣ Gn s ♣rs♥t ♥ r

t♦r r♣ Gn s s♣② ♥ t ♥tr r♠ ss♦t♣rs♦♥ ♥t♦r ss♦t t♦ ♥♦ s s♣② ♥ r♠ t r♦♥

♥ Prs♦♥ t♦rs

②s ss♠♥ t s♠ ♥♦tt♦♥ ♦♥♥t♦♥ t st ♦ ♥♦s ♥ t st ♦ s♦ Gk

p r s♥t ② V kp ♥ Ek

p rs♣t② ② ♥♦t♥ Vn(Rk ) t ♥♦s ♦

Gn sr♥ t s♠ st ♦ Rk t ♥♦s V k

n ♦ ♣rs♦♥ ♥t♦r Gkn ♥

rtt♥

V kp = Vn(R

k )

② ♦♥strt♦♥ ♥♦ k ♦ t ♣rs♦♥ ♥t♦r Gkp s ♥ t♦ t ♥♦s ♦ ts

♣rs♦♥ ♥t♦r ♠r② t♦ Gn ts ♥ s ♦♥② ♣♦t♥t ♦♥♥t♦♥ ♥ ♦ ♥♦t♥♦r♠ ♦♥ t t ♦♥♥tt② ♦ t ♥♦ t ts ♥♦rs tt ♥♦r♠t♦♥ s♠♥ ② ♦♦♥ ♦♥ t ♥ ♥t♦ ♦♥srt♦♥ tt ♣rs♦♥

♠t♦♥ ♦ t ♦ t♦r r♥③t♦♥

♥t♦r s rt♦♥ r♣ t st ♦ s ♥ rtt♥

Ekp =

⋃

l⊂V kp

l 6=k

el,kn

⋃

⋃

m⊂V kp

m 6=k

ek,mn

r el,kn ♥ ek,mn r s ♦ Gn ♥♥ ♥♦s l k ♥ k m rs♣t②

♦s ♦ Prs♦♥ t♦r

♥ ♦rr t♦ ♣r♦r♠ rst s♠t♦♥ t ttrts ♦ t ♣rs♦♥ ♥t♦rs♥♦s r ♠t t♦ ♥♦r♠t♦♥ s♣♣♦s t♦ ② t ♥♦ ♥ r s

ttrt ♦ ♥♦ V k

p s ♦t♥ r♦♠ t s♠ ♥♦ V kn ♦ Gn

st♠t ♣♦st♦♥ st♠t ♣♦st♦♥ s ♦t♥ r♦♠ t ♦♠♣tt♦♥ ♦ ♦③t♦♥ ♦rt♠ tr r② ♦r ♦♠tr② s♥ ♦r ♥st♥ t P ♦rt♠sr ♥ st♦♥

♠ st♠♣ ♦ st♠t ♣♦st♦♥s ♦ t sr♣t♦♥ t ♣♦st♦♥ st♠t♦♥ s s②♥r♦♥♦s② ♣t st♠t♦♥ s t♦ t♠ st♠♣ ♦r t s ♦ ♣♦st ♣r♦ss♥

s ♦ Prs♦♥ t♦r

♣rs♦♥ ♥t♦r Gkp s ♠tr♣ r ♦rrs♣♦♥s t♦ r♥t

♦ t ♣rs♦♥ ♥t♦r ♦♥t♥s ttrts ♦t t ♦♥♥t♦♥s♦ k t t ♦tr ♥♦s ♦ ts ♥♦r♦♦

P t♦♥r② s ♦ t P t♦♥r② r Ps ♥ rr ♦ st♥rs♦s s r rt② ♣t r♦♠ t♦s ♦♥ Gn s t s♣rrs t♠ s ♣r♦r ♠s t♦ ♥rs t rs♠ ♦ t s♠t♦♥ ②♠t♥ t ♠♦♥t ♦ ♥♦r♠t♦♥ ② ♥♦ rrs t♠ ♦ ♣rs♦♥ ♥t♦rs P s rqst ② t ♦③t♦♥ st s ♣rtr♣r♦ss s t t ♥①t sst♦♥

P t♠ st♠♣ t♦ t sr♣t♦♥ P ♥♦r♠t♦♥ s t♠ st♠♣ ♦r t s♦ ♣♦st ♣r♦ss♥

❱st② st② t♥ t♦ ♥♦s k ♥ l s ♣t ② t♥ ♦♥t t s♥stt② ♦ ♥♦ k t ♠tt ♣♦r ♦ ♥♦ l ♥ t ♦♠♣t r ♣♦rP ♦ t♦s t♦ ♥♦s

♦t tt t q♥tt② ♦ ♥♦r♠t♦♥ ♥ ♦t t ♥♦s ♥ t s ♦t ♣rs♦♥ ♥t♦r ♦ ♥t♦s② ♥rs r s st t ♠♥♠ ♣♦ ♥♦r♠t♦♥ rqr ♦r t ♣♦♠♥ ♦♠♠♥t♦♥ st

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

♠t♦♥ ♦ t ♦♠♠♥t♦♥ t

Ps ♥♦r♠t♦♥ ♦♥t♥ ♥ t s ♦ t ♣rs♦♥ ♥t♦r r rrst r♥t rt tt t♦s ♦♥ t s ♦ t ♥t♦r ♥ ♣rs♦♥ ♥t♦rss r rrs ♦♥② ♦♥ t rqst ♦ t ♦③t♦♥ st ♥ ♥♦ ♥♦ Pr ♦r tt ♣r♣♦s t ♦♠♠♥t♦♥ st s ♥tr♦

♦♠♠♥t♦♥ t Pr♦ss♥

♦♠♠♥t♦♥ st ♥s t ♥♦s ♦ ♥ ♣rs♦♥ ♥t♦r ♣r♦ss♥ ♦ ts st s ♥♣♥♥t ♥ ♥q ♦r ♥t s st s r♥② t ♦③t♦♥ st ♥ t ♦③t♦♥ st ♦ ♥ ♥t tts tt trs ♦ P t♦ st♠t t ♣♦st♦♥ ♦r tt t ♥♦r♠t♦♥ r ♦s♦t rrsrqst s s♥t t♦ t ♦♠♠♥t♦♥ st

♦♠♠♥t♦♥ st ♦ ♥t s ♥ t♦ ♣r♦sss r♣t♦♥♣r♦ss t♥ ♦r rqst ♥ tr♥s♠ss♦♥ ♣r♦ss s♥♥ rqsts t♦ ♦tr♥ts r♣t♦♥ ♣r♦sss tr♥s♠ss♦♥ ♣r♦ss s trr ② ♦③t♦♥st t t ♣ ♦ rrs rqst ♥ ♥♦ k r rrs rqstts tr♥s♠ss♦♥ st s♥s s♣ rqst t♦ t r♣t♦♥ st ♦ ♦ ♣rt ♦ t♥♦s ♣♥♥ ♦ t ♦s♥ ♦♥rt♦♥ ❲♥ t r♣t♦♥ ♣r♦ss r rqst r♦♠ tr♥s♠ss♦♥ st ♦ ♥♦ k s♣ ♣r♦♥ s ♣r♦r♠ t♦♣t t ♣rs♦♥ ♥t♦r ♦ t ♥♦ k t s ♦♥t♥s ♦ t ♣rs♦♥ ♥t♦r♦ t rqst ♥♦

Prt② t rqsts r ♥rt s♥ ♠P② ♠♥ts ❬❪ ♠♥ts♦ t♦ tt ♦r ♥trr♣t ♠P② ♣r♦ss ①t♦♥ t t ♣ ♦ s♥ s♥ ♦ t ♦♠♠♥t♦♥ st ♦rrs♣♦♥s t♦ s♣ rqst ♦t r♦♠ ♥♥♦ t♦ ♥♦tr ♦♥ ♥ ♦♥ ♥ ♦r ♦♠♣tt♦♥ ♦♥♥♥ t♦s s♥sr st♦r s s ♦ ♦♠♠♥t♦♥ t♦♥r② r t ②s r s♠♣ ♠ss(k, l, r) s♥t♥ t ♦ t rqstr ♥♦ t ♦ t rqst ♥♦ ♥ t ♦ t ♦♠♠♥t♦♥ rs♣t② t t♦♥r② sr t♥ ♥♦s ♦Gn ♦♥t♥s t ♣♦ss ♥tr♥♦s s♥s ♦♠♥t♦♥s

ts ♦♥sr t ♣rtr stt♦♥ ♦ r r t ♦③t♦♥ st ♦ ♥♦ k s ♥♦t ♥♦ Ps t♦ st♠t ♣♦st♦♥ ♦③t♦♥ st s♥ rrs rqst t♦ t ♦♠♠♥t♦♥ st ♥ t tr♥s♠ss♦♥ ♣r♦ss ♦ ♥♦k s♥s rqst t♦ t ♥♦s l ♥ m ♦♥♥ t♦ ts ♣rs♦♥ ♥t♦r ♦r tt♣r♣♦s t ss ♠sss ♦ t ♦♠♠♥t♦♥ t♦♥r② (k, l, r) ♥ (k,m, r) t♦rqst ♥♦r♠t♦♥ ♦ ♥♦s l ♥ m ♦♥ r r♣t♦♥ ♣r♦sss ♦ ♥♦s l♥ m r trr ② t s♥s ss♦t t♦ t t♦ ♠sss (k, l, r) ♥ (k,m, r)rs♣t② ♥② rr ♣r♦sss ♦ l ♥ m ♣t t ♣rs♦♥ ♥t♦r ♦ ♥♦k t t ♥♦r♠t♦♥ ♦♥t♥ ♥ tr ♣rs♦♥ ♥t♦rs

t ❯♣t

♦♠♠♥t♦♥ st ♣r♦ss♥ ♦s t♦ ♣t ♥♦r♠t♦♥ t♥ ♥♦s♦ ♥ ♣rs♦♥ ♥t♦r t♦ ♥♦tr ♥ rst ♣♣r♦ tt ♣r♦r s s t♦

♠t♦♥ ♦ t ♦♠♠♥t♦♥ t ①♠♣ ♦ ♦♠♠♥t♦♥ ♣r♦ss ♥tt ② t ♦③t♦♥ st ♦ ♥♦k s ♥♦ ♥♦r♠t♦♥ r t♦ st♠t ts ♣♦st♦♥ t ♦③t♦♥ st♦ ♥♦ k rqsts ♦♠♠♥t♦♥ tr♥s♠ss♦♥ st ♦ ♥♦ k s♥s rqstt♦ ♥♦ l ♥ m ♦t ♥♦s ts ♣t t ♣rs♦♥ ♥t♦r ♦ ♥♦ k

♣t r♦ ♦srs r♦♠ ♥♦ t♦ ♥♦tr Prt② t ♠♣♠♥tt♦♥ ♦tt ♣r♦ss♥ ♦ ♥♦t t② rqrs ♥② t ①♥s t♥ ♥♦s t ♦♥②rqst ①♥s t ♣t s ♣r♦r♠ ② t ♥♦ r♥ t rqsts♥ t s r♦♠ ts ♣rs♦♥ ♥t♦r ♥ ♥♦ s t♦ sr ♥② t②♣♦ ♥♦r♠t♦♥ ♦♥t♥ ♥ ts ♣rs♦♥ ♥t♦r ♠♦♥

ts ♦srs Ps ts st♠t ♣♦st♦♥ ts ♦t② ♥ rt♦♥ t♦rs ts st② ♥♦r♠t♦♥

①♥♥ t Ps ♥♦r♠t♦♥ ♣s t♦ s♠t ss t ①♥ r♥ ♦♠♠♥t♦♥ st♠t ♣♦st♦♥ ♥ t st② ♦s t♦ rss ♦♦♣rts ♣♣r♦s r t ♣♦st♦♥ ♦ ♦tr r♦ s ♥ ①♣♦t t♦ st♠tt ♥♦ ♣♦st♦♥ ❬❪ ♦t② ♥ rt♦♥ t♦r ♥s ♦♦♣rt ts♦♥ ♦rt♠

s ♠♥t♦♥ ♦r t ♥♦r♠t♦♥ ♦♥t♥ ♥ t ♣rs♦♥ ♥t♦r ♦ ♥♦ ♥ s② ①t♥ t ♥② t②♣ ♦ ♥♦r♠t♦♥ ♦s ①tr♥♦r♠t♦♥♦ s♦ sr ② t ♦♠♠♥t♦♥ st ♥ ♥♦ t♥ ts ♣♦st♦♥t t ♣ ♦ ♠ss ♣ss♥ ♠t♦s s sr ♥ ❬❪ ♦ s② srt P ♥♦r♠t♦♥ t ts ♣rs

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

♦♠♠♥t♦♥ strt♦♥

♥♦tr s♣t ♦ t ♣r♦♣♦s ♦♠♠♥t♦♥ st ♦♥ssts ♥ ♣♣②♥ s♦♠ rst♦ rstrt t t ①♥s ♥ ♦♦♣rt s♠s ts tr ♦s ♥♦ ♥♦tt♦ ♣r♦ s♣ ♣rs♦♥ ♥t♦r t ♦♥ rqst r♦♠ ♦♥ ♣r ②♣② tttr ♥ s ♥ ♦♦♣rt ♦rt♠s t♦ tst ♥s♦r♥ t ♦♥ t ♣♦st♦♥st♠t♦♥ ❬❪ ♦r ♥ ♥ st♦♥ t♦ st♠t t ♥♥ ♦ t ♦ t♦♥ ♦Prs♦♥ P ❬❪ ♥ ♥ sr ♥ st♣ t ♦♠♠♥t♦♥ rstrt♦♥♦♥ rtr ♦

r ♣♦r st♥ P ♠♦s st♠t ♣♦st♦♥ ♠t♠♦ s♦t♦♥s ♣r♦ ② P

♦s② ts st ♦ ①t♥ ♥ t tr t♦ rss ♠♦r ♦♠♣① s♥r♦s

①♠♣ ♦ ❯s ♦ t ②♥♠ Pt♦r♠

s st♦♥ ♣r♦♣♦ss t♦ strt s♦♠ ①♠♣s ♦♥ ♦ t ②♥♠ s♠t♦♥t ♣r♦♣t♦♥ t♦♦s ♥ t ♦③t♦♥ ♦rt♠s ♥ s t♦tr

rst s♥r♦ sr ♥ ♦♥srs ♥ ♥t ♠♦♥ ♥t♦ t ♥r♦♥♠♥tr t s ♦♥♥t t♦ ♥♦rs ♥♦s t s♥r♦ s s ♥ t♦ r♥ t♠t ♦rt♠ t♦ ♣r♦ rr② ♦ ♥ Ps t♥ t ♥t ♥t ♥♦rs ♥♦s s ♦♥ t s♠ s♠t♦♥ t s♦♠ r ♦♠♣t s♥t t♦♦ ♥ ♥② s♦♥ s♠t♦♥ ♥♦♥ ♠♦ ♥ts s ♣r♦♣♦s♥ t♦ ♠♦♥strt t ♦♠♠♥t♦♥ st

s♣②♥ t ♦t②

①♠♣ strt ♥ r s ♦♥sr t♦ t t ♠t ♣♣r♦ ♥ t t♦♦ ♥ ts ①♠♣ s♥ ♥t s ♠♦♥ ♥t♦ t ②♦t ♥ s♥ st② ♦ ♥♦rs ♥♦s tr ♥♦rs ♥♦s r r♣rs♥t t tr♥s r♥t r♦♥ trt ♣♦st♦♥s ♦ t ♥t r s♠♣ r② s♦♥ ♥ ♣♦tts r ♦ts t♥ ♦ts 1 s♦♥ s ♣s r② 10 s♦♥s t s♠ ♦tsr r♣ ② rr ♦ts t ♥ ♥① r t ♦s rr ♦ts r ♣♦♥t♣♦st♦♥s ♥ t② ♦ t♦ tr♠♥ t rt♦♥ ♦ t ♥t ♥ t♦ rr♥♣♦♥ts t♦ s② ♠t t ♣♦st♦♥ ♥ t P s ♥ t ♦♦♥ rs

r♦♠ tt s♠t♦♥ ♦ s s ♦♥t♥♥ t ♣♦st♦♥s ♦ ♥♦s♠♦s ♥ stts ♥ t r♦ ♥ t♥ ♥♦s ♥ t ♦ ♦♥t♥s

tr ♥♦ ♣♦st♦♥s st♠t ♥♦ ♣♦st♦♥s ♦♥ ♦③t♦♥ ♦rt♠ s ♥ st♣ r ♣♦r ♥ ② ♦r t ♥s t t♠st♠♣

①♠♣ ♦ ❯s ♦ t ②♥♠ Pt♦r♠

30 20 10 0 102

4

6

8

10

12

14

16

18

agent #1

AP #1

AP #2

AP #3

t0t1 t2

♠t♦♥ ♦ trt♦r② ♦ ♠♦ ♥t t stt ss ♣♦♥ts

①♣♦t♥ t ♦t♦♥ ♣♥♥t Pr♠tr P ❯s♥ tt❲ ♣♣r♦

r♦♠ t trt♦r② t s ♣♦ss t♦ t t ♦t♦♥ ♦ Ps t♥ t r♦♥s r s♦s t ♦t♦♥ ♦ ♦t t r ♣♦r ♥ t ①ss t♠♦ ② ♦♥ t ♠♦ ♥t trt♦r② r♥t ♣♦ts ♦rrs♣♦♥ t♦ r♦♥s ♥♦♥ t ♠♦ ♥t ♥ t stt ss ♣♦♥ts rs♣t② ♦②♥ ♥rs ♦ t r ♣♦r ♦rrs♣♦♥s t♦ rs ♦ t ②

♦♠♣t♥ t ❯s♥ t

♥r♠♥t t♦♥ ♦ t r♥ t ②♥♠ s♠t♦♥ s ♥♦t ♥② ♠♣♠♥t ②t ♦r t s r② ♣♦ss t♦ s t s♠t♦♥ ♦ t♦ ♣♦st♣r♦ss t ②♥♠ s♠t♦♥ t t t♦♦ ♥s t t ♦ t ♥s♦ t ②♦t

♥ ♦ t ♥t ♦ t r♣ s s t s♥tr sr♣t♦♥ s ①♣♥ ♥ ♣tr s♥trs ♥ ♥tr♦ t♦ r♣rs♥t t ♣rsst♥t♦♠♣♦♥♥t ♦ t r♦ ♥♥ t ♠♥s tt ♦r ♥ ♥r♦♥♠♥t t ♥♥♣rtrt♦♥ r s ② s♦♠ s♣ s♥trs ♥ rt② rt t♦s♣ ♣ttr♥ ♥t♦ t

t♥ r♦♦♠s s♥trs r ss♠ t♦ stt♦♥r② Prt② t ♠♥stt t tr♥s♠ttr ♦r t rr r♠♥ ♥ tr rs♣t r♦♦♠s t s♠st ♦ s♥trs ♥ s t ♥ ♥♦t tt t s♥tr sr♣t♦♥ s ♦♥②rt t♦ t ♥r♦♥♠♥t ♥ ♦ ♥♦t ♥♦ t ♣♦st♦♥ ♦ ♥tr t rr ♥♦rt tr♥s♠ttr ♦♥sq♥t② s ♦♥ s ♥ ♥t ♦♥ t♦ s♠ r♦♦♠ t s♠st ♦ s♥trs ♥ rs t♦ t ss♦t r②s

s ♥ strt♦♥ s ♦ t ♠♦ ♥t s ♦♠♣t ♦r ♣♦♥ts ♦ ts trt♦r②♦♥♥ t♦ t s♠ r♦♦♠ ♥ t t ♦♠♥ ♦ r r s♣② t♦t♥ r②s t♥ ♣♦st♦♥s ♦ t ♥t ♥ ♥ ss ♣♦♥t ♥ ts ②♦tr♣rs♥tt♦♥s t ♦rrs♣♦♥♥ ♦t♥ r s♣② st ♦ r②s ♥

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

2 0 2 4 6 8 10 12time (s)

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we

r(d

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2 0 2 4 6 8 10 12time (s)

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we

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2 0 2 4 6 8 10 12time (s)

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lay

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Rece ived Power

Delay

t0 t1 t2

link agent #1 and AP #2



t0 t1 t2

t0 t1 t2

rs ♦rrs♣♦♥ t♦ r♥t r♦ ♥s ♥ r r ♣♦tst♥♦s② t r ♣♦r ♥ t st♠t ② s ♥t♦♥ ♦ t s♠t♦♥t♠ ♦t ♦rrs♣♦♥s t♦ ♥ st♠t♦♥ ♦ r ♣♦r rs r♦ss♦rrs♣♦♥ t♦ ♥ st♠t♦♥ ♦ t ②

①♠♣ ♦ ❯s ♦ t ②♥♠ Pt♦r♠

s ♥ ♦t♥ s♥ t s♠ st ♦ s♥trs♦r t s ♦t♥ ♥ r r s♣②s ♥ st♦r♠ ♦ t

♦♠♣tt♦♥ t♠ ♦r ♦ t♦s s♥trs s ①♣t t ♦♠♣tt♦♥ ♦ trst s ♦♥r t♦ t ♦ strt ♦♠♣tt♦♥ ♦ t s♥trs ♦r ♦rt ♥①t ♣♦st♦♥s ♦ t ♠♦ ♥t st ♦♥ t♦ t s♠ ② sr ♦t♥ s♥♥t② ♠♦r r♣②

s ♥ ①♠♣ r s♦s ♦♠♣rs♦♥ ♦ s♠t t♦ ♠sr♠♥t♦t♥ r♥ t ♠sr♠♥t ♠♣♥ ♦ t ❲ ♣r♦t ❬❪ ♦r ts①♠♣ t ♠ss♦♥ ♣♦r ♦ t tr♥s♠ttr s ♥♦♥ ♥ s ♥ s t♦ s t t♦♦ s♠t♦♥ ♦s ♥♦t ♥ t t ♥♦s ♥ t ♥ ♦srtt t ② ♦ ♦t t rst ♣t t 27 ns ♥ ♥♦tr ♦♥trt♦♥ t 36 ns r♦rrt② st♠t ② t t♦♦ ♦r t s♠t♦♥ ♣♥s ♦♥ t sr♣t♦♥♦ t ♥r♦♥♠♥t ♦r tt s♠t♦♥ r♥trs ♥♦t ♥ t♥ ♥t♦ ♦♥ts ♥ ①♣♥ ② str♦♥ ♦♥trt♦♥ ♥ ♦sr ♥ s♠t♦♥ t 30ns ♥ ♥t♦ t ♠sr t ♥ t ♦♥trr② sr ♦♥trt♦rs ♣♣r♥t♥ 32 ♥ 35 ns ♦♥ ♠sr♠♥t ♥ r ♥♦t rtr ② t s♠t♦♥ sr♥ ♥ ①♣♥ ② ♠t ♥♠r ♦ r②s rsr r♥ t s♠t♦♥♦r t s♠t♦r s s♥ ♦r ♦③t♦♥ ♣r♣♦s ♠ r s ♥♦tt♦ t t ♥♥ ♦♥trt♦rs t t ♠♦st s♥♥t ♦♥ r rqr t♦♣r♦ rt s♠t Ps ♥ ts ♦③t♦♥ ♦rt♠s ♦r ①♠♣s♦ t t♦♦ r ♣r♦ ♥ t ♥②③ ♦ t ❲ ♠sr♠♥t ♠♣♥ ♥♣tr

①♠♣ ♦ ❯s ♦ t ♦♠♠♥t♦♥ t

s♠t♦♥ r♣rs♥t ♥ r s ♦♥sr t♦ ♠♦♥strt t ♦♠♠♥t♦♥ st r ♠♦♥ ♥ts ♥ stt ♥♦rs r ♦♥sr t♦♠♦ ♥ts strt r♦♠ r♥t s ♦ t ②♦t ♠♦ t♦r ♦tr r♦sst♠ss ♥t♦ ♦rr♦r ♥ ♦♥t♥ t♦ tr st♥t♦♥s r♥t trt♦rs ♦ t ♥t 1 ♥ t ♥t 2 r r♣rs♥t ② r ♥ t ♦ts rs♣t②♠r② t♦ t ♣r♦s s♠t♦♥ t♥ ♦ts 1 s s ♣s ♥ ♣♦♥t♣♦st♦♥ s st r② 10 s♦♥s ♦♥sr♥ t ♥t♦r ♦r♥③t♦♥ 3 r♥t sr ♥♦ ♥ t s♠t♦♥ rat1 rat2 ♥ rat3

♠♦ ♥t 1 ♠s rat1 ♥ rat3 ♥ t ♠♦ ♥t 2 ♠s rat2♥ rat3 s s t t♦ ♠♦ ♥ts ♥ ♦♥② ♦♥♥t t♦tr ♦♥ rat3 ♠♦ ♥t 1 s ♦♥♥t ♦♥ rat1 t♦ t stt ♥♦rs r♣rs♥t t tr tr♥s rs t ♠♦ ♥t 2 s ♦♥♥t ♦♥ rat2 t♦ t stt ♥♦rsr♣rs♥t t t t tr♥s ♥② s♦♥ ♥t trs t♦ st♠tts ♣♦st♦♥ t t ♦ ♦ ts ♣rs♦♥ ♥t♦r ♥♦t ♥♦ ♦ r t tr② t♦ ♦♠♠♥t t ts ♥♦r♦♦ ♦♠♠♥t♦♥s st s♥ ♦♥r t♦ ①♥ ♦t ♦ ♥ st♠t ♣♦st♦♥s

r r♣rs♥ts t ♣♦st♦♥♥ rr♦r ♦ t ♠♦ ♥t s ♥t♦♥ ♦ tt♠ rst ♦ t ③r♦ s t t ♥♥♥ ♦rrs♣♦♥ t♦ ♥ ♥t③t♦♥ ♥ st ♠♥ tt t ♥ts ♥t ②t strt t♦ st♠t t ♣♦st♦♥ ♥ ♦t

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

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rs ♦♥ t t s♦ ♦♠♣t r②s t♥ t ♠♦ ♥t ♥♥ ss ♣♦♥t ♦r r♥t ♣♦st♦♥ ♦ t ♠♦ ♥t ♦♥ t trt♦r② ♥ trt ♦♠♥ r s♣② t t s ♦rrs♣♦♥♥ t♦ t stt♦♥ ♦♥ tt ♦♠♥

①♠♣ ♦ ❯s ♦ t ②♥♠ Pt♦r♠

1 2 3 4 5 6trajectory point number

0

1

2

3

4

5

6ti

me (

s)

gain:4.40 x

gain:4.40 x

gain:4.54 x

gain:4.55 x

gain:4.78 x

execution time as a function of the trajectory point

♠ rqr t♦ ♣r♦ ♦r r♥t ♣♦st♦♥s ♦ t ♠♦ ♥♦ ♥ ♥t♦♥ ♥♦r♠s ♦t s♣ ♠♣r♦♠♥t ♥ ♦♠♣tt♦♥ rt t♦t rst ♦ strt rqr♥ s♥tr ♦♠♣tt♦♥ ♣♦st♦♥ s ♦♠♣tr t♦♣r♦r♠ tt s♠t♦♥ s ♥ ♥t ♦r t ♦

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80Time (ns)

0.0020

0.0015

0.0010

0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

Volt

age (

V)

Measurements

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80Time (ns)

0.0020

0.0015

0.0010

0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

Volt

age (

V)

Simulation-with antenna

♥ ①♠♣ ♦ ♦♠♣tt♦♥ ♥ s s♦♥ t stt♦♥ ♦ t s♠t♦♥ ♥ t r②s ♥rt ② t r② tr♥ ♦♠♣tt♦♥ ♥ t s♠t♥ ♠sr s ♦ t stt♦♥ r r♣rs♥t s ♥t♦♥ ♦ ②

♣tr ②♥♠ Pt♦r♠ ♥♥ Pr♦♣t♦♥ ♦♠♠♥t♦♥ ♥ ♥t

♦t② ♥ t t♦r

30 20 10 0 104

6

8

10

12

14

16 agent #1

agent #2

AP #2_1

AP #1_1

AP #1_2

AP #2_2

t0t0

t1 t2 t0t1t2

♠t♦♥ s♥r♦ r♦♥ trt ♣♦st♦♥s ♦ t ♠♦ ♥t 1 ♥ ♠♦♥t 2 r r♣rs♥t ② r ♥ t ♦ts rs♣t② ♥♦rs ♥♦s ♦ ♠♦♥t 1 ♥ ♠♦ ♥ts 2 r r♣rs♥t ② r ♥ t tr♥s rs♣t② ♥t♦♥ ♦t ♠♦ ♥ts ♥ ♦♥♥t t♦tr

0 2 4 6 8 10

time (s)

0

1

2

3

4

5

6

7

err

or

(m)

0 2 4 6 8 10

time (s)

0

1

2

3

4

5

6

7

err

or

(m)

agent #1

agent #2

t0 t1 t2

t0 t1 t2

P♦st♦♥♥ rr♦r ♦ t ♠♦ ♥ts s ♥t♦♥ ♦ t t♠ ♣r♦①♠t② ♦ t t♦ ♥ts ♥♥ t ♦♠♠♥t♦♥ ♣r♦ss s ♦sr t♥4s ♥ 7s

♦♥s♦♥

rs r♠s ♥ ♦sr r♠ ♦ r ♣♦st♦♥♥ rr♦r ♦rrs♣♦♥♥t♦ stt♦♥ r ♠♦ ♥ts ♦♥② ss t♦ ♥♦rs ♥♦s ♦r ♣♦st♦♥♥♥ r♠ ♦ s♠ ♣♦st♦♥♥ rr♦r r tr ♥♦r t ♦tr ♥t ♣s t♦st♠t t ♣♦st♦♥

♥ t s ♥♦t tt s♦♥t♥t② ♣♣rs ♥ t ♣♦st♦♥♥ rr♦r t 5s♦r ♥t 1 ♥ 4s ♦r ♥t 2 ♦♥sr♥ t ♣♦♥t s ♦rrs♣♦♥st♦ t♦s r♣rs♥t ♥ r t ♣♣rs tt t rs ♦ t ♣♦st♦♥♥rr♦r ♦rrs♣♦♥s t♦ stt♦♥ r t ♠♦ ♥ts r ♥ st② ♥ tt♦ ♥ts ♦♠♠♥t t♥ ♦tr ♥ ♦rr t♦ ①♥ ♦t tr st♠t♣♦st♦♥ ♥ ♦ s ♦♥ s t② r ♥ st② t ♣♦st♦♥ rr♦r r♠♥s ♦ t t♠ 7 s♦♥s ♦t ♥♦ ♦♦s tr rs♣t st② ♥ ts t♣♦sst② t♦ ①♥ ♥♦r♠t♦♥ ♥ t ♣♦st♦♥ st♠t♦♥ s ♥♦ ♦♥r rt♥ t ♣♦st♦♥♥ rr♦r ♥rss ♥

♦♥s♦♥

t t ♥♥♥ ♦ ts ♣tr stt ♦ t rt ♦♥ t ♠♦t② ♠♦ ♥ ♦♥ t♥t♦r s♠t♦r s ♥ ♣r♦ ♥①t st♦♥ t t ♥t ♠♦t②♥t♦ t ♣r♦♣♦s ②♥♠ ♣t♦r♠ t ♠♦t② s ♥♠t ② ♥ ♥ s♣t ♥t♦ t♦ ♣rts rst ♣rt t♥ t r s ♠♦t② s ♦♥ t r♣sr♣t♦♥ s♦♥ ♣rt t♥ t s♠ s ♠♦t② ♦ t ♥t s s♦♥♣rt s ♦♥ t str♥ ♦r ♠t♦ s sr ♦ ♥ ♥t ♥ ♠♦ ♥t♦ ♥ ♥r♦♥♠♥t ♥ ♥trt t ♦tr ♥ts ♠♥ ♦r♥ ♣rt ♦ tt♦r s t ♣tt♦♥ ♦ t str♥ ♦r ♠t♦ t♦ t r♣ r♣rs♥tt♦♥

s ♣tr s s♦ ♥tr♦ sr♣t♦♥ ♦ t ♦♣t r♣♦r♥t ♥t♦r♦r♥③t♦♥ ♥ ♣rtr ts st♦♥ s t ♦ t ♥ts ♦ t trPs ♦t♥ ② t ♠t ♠t♦ sr ♥ t rst ♣tr s♦♠♠♥t♦♥ ♣r♦t♦♦ s s♦ ♥ sr ♦♥ t ①♥ ♦ ♠ssst♥ ♥ts

♥② t st st♦♥ s ♣r♦ t♦ ①♠♣s ♦ s♠t♦♥s rst s♠t♦♥ s s♦♥ s♥ ♥t ♠♦♥ ♥t♦ s②♥tt ♥r♦♥♠♥t r Ps ♥ ♥ t r♦♠ ts trt♦r② s♦♥ s♠t♦♥ s ♠♦♥strt♦ t ♦♠♠♥t♦♥ st ♦ s ♦r rss♥ ♦♦♣rt ♦③t♦♥♣r♦♠s

♣tr

t♦♥ ♦ t Pr♦♣♦st♦s ♦♥ sr t

♥tr♦t♦♥

s ♣tr ♣r♦♣♦ss t t♦♥ ♦ t ♠t♦s ♣rs♥t ♥ t ♣r♦s ♣trs s ♦♥ t t ♦t♥ r♦♠ ♠sr♠♥t ♠♣♥ r③ r♥ r♦♣♥♣r♦t P ❲

rst st♦♥ srs t tr♦♥♦s ♣t♦r♠ s t♦ ♣r♦r♠ t ♠sr♠♥t ♠♣♥ r♥t s r♦ t♥♦♦② r st ♥ t r♥t♠sr♠♥t s♥r♦s r t ♥ t ①trt♦♥ ♦ ♥♥ ♣r♠trs s♣r♦r♠ ♦♥ s♣ s♥r♦ ♦ t P st ♠sr♠♥t ♠♣♥

s♦♥ st♦♥ ♣r♦s ♦♠♣rs♦♥ t♥ ♠sr Ps ♥ s♠t Ps s♥ t ♦♣ s♠t♦r ♥ rst t♠ ♦♠♣rs♦♥ s ♣r♦s♥ t st s♣ ♠t t♦ t Ps ♦♥ ♠sr ♣♦♥ts ♥ s♦♥t♠ st s♠♣ ♦ ♠sr r ♦♠♣r t♦ t♦s ♦t♥ t t t♦♦

tr st♦♥ s ♥ tt♠♣t t♦ t t ♦③t♦♥ ♦rt♠s ♣rs♥t♥ ts tss s ♦♥ ♦t s♥r♦s ①trt r♦♠ t P st ♠sr♠♥t ♠♣♥ ♥ t st♠t ♣♦st♦♥ r② s t s♥ t ♣r♦♣♦s P♠t♦ ♥ ♦♠♣r t♦ st♥r ♣♣r♦ ♦r ♠♦ ♣♦st♦♥s ♦ t tst ♥② t ♥trst ♦ ♦③t♦♥ ♠t♦ s ♦♥ ②♣♦tss tst♥ s ♠♦♥strt ♠♥ s ♦ st♦♥ ♦ ♠sr ♦s ♥ ♦♠♣♠♥tr② ♠sr

sr♣t♦♥ ♦ t sr♠♥t ♠♣♥

s st♦♥ ♦ss ♦♥ t sr♣t♦♥ ♦ t ♠sr t ♦t♥ r♥ tP ❲ Pr♦t rst ♦ sr♣t♦♥ ♦ t ♠sr♠♥t ♠♣♥ s♣r♦♣♦s ♥ t ♥♥ ♣r♠trs r ①trt r♦♠ ♥ s♥r♦

♣tr t♦♥ ♦ t Pr♦♣♦s t♦s ♦♥ sr t

P ❲ Pr♦t

s tss s ♥ r③ ♦♥ t r♦♣♥ P ❲ Pr♦t ❬❪ ❲ ♣r♦t ♠s t♦ ♥♥ t ♥② ♦ tr rss ♦♠♠♥t♦♥ss②st♠s ② s♥ ♦t t ♣♦st♦♥♥ ♥ ♦♠♠♥t♦♥s ♥♦r♠t♦♥ st♠t♦♥♦ t ♣♦st♦♥ ♦ ♠♦ tr♠♥s ♦t♥ s♥ tr♦♥♦s ♦③t♦♥♦rt♠s s♥ r♥t t②♣s ♦ r♦ ♦srs ♦t♥ r♦♠ r♥t s

♣ts ♦ t P sr♠♥t ♠♣♥

❲ ♠sr♠♥t ♠♣♥ s ♥ r③ r♥ ②s ♥ ♣r ♥ t ♦ ♦ P ♣rt♥r ♥ r♦ P♦rt rsts ♦ ts ♠sr♠♥t♠♣♥ r ♣ ♥ ♥ ♦♥♦ r♦♠ ❬❪ t rt r♦ss t♥♦♦s ♥ s ♦r t rst t♠ ♦♥t② ♣r♦♥ r② ♥qtr♦♥♦s r♦ tst ♦r t♥ ♣♦st♦♥♥ ♣♣t♦♥s ♣t♦r♠ s♦♠♣♦s ♦

s♦rt r♥ r♦s ❩ ❯❲ ♦♥ r♥ r r♦ PP sr ♠ ♥rt s♥s♦rs r♦♠tr ♠♥t♦♠tr ②r♦♠trr♥ t ♠sr♠♥t ♠♣♥ r♦ ♥ t ss♦t t♦ r♦

t♥♦♦② ♥ st♦r ♥t♦ t♠ st♠♣ t s ♠sr♠♥t tsts ♥ ♠ ♦♥ ♠♦ tr♦② s ♥ ♠♦ ♦♥ t P ♥

♠sr♠♥t tst s q♣♣ t ♦ t②♣ ♦ s♦rt ♥♠♠ r♥ r♦ ♣t♦r♠ ♥ ♥rt s♥s♦rs ♣♥♥ ♦♥ t s♥r♦s tr♦♠ tr ♦r st ♥♠r ♦ s ♥ st♦r ♥ ts ♦r ♣♦st♣r♦ss♥

trt♦rs ♦ t tr♦② ♥ ♣rtr♠♥ ♥ r s♣ ♦r s♥r♦ s r♦ s ♥ t♦ tr ♦♦r♥t♦rs ♦♥ t tr♦② ♥s♣r ♥t♦ t ♠sr♠♥t r ♥ t P ♥ ♦♠ ♦ t♦s r♦ sr ♠♦ rs ♠♦st r stt ss ♣♦♥ts

r s♦s r♣rs♥tt♦♥ ♦ t ❲ tst ♠ ♦♥ t tr♦②♥ t ♦r♥③t♦♥ ♦ t s ♦ s r♦♥

♦♥sr ♦ ♥♦♦s

s ♦r s ♥ ♦s ♦♥ t ①♣♦tt♦♥ ♦ s♦rt r♥s r♦ t♥♦♦s♣r♦♥ t②♣s ♦ r♦ ♦srs

❩ ♣r♦s ♦srs ♥♥ ♥r② Pr♦ P r♦♠ t ❯❲ ♣t♦r♠ ❬❪ ♥ trss♦t ♦ st♠ts

❩ ♦s

❩ ♥♦s r ♦♠♣♥t tr♥sr s♥ ♦r ♦♣♦r ♥ ♦♦t rss ♣♣t♦♥s ♠♥ s♣t♦♥s ♦ t s r

sr♣t♦♥ ♦ t sr♠♥t ♠♣♥

❲r ♠sr♠♥t ♣t♦r♠ r♥t r♦ t♥♦♦s ❩❯❲ r ♠ ♦♥ ♠♦ r♦②

❬❪ rq♥② GHz ♥ ♥t MHz ♦t♦♥ P ① s♥stt② dBm

s r ♦r♥③ ♥ ♥tr③ t♦♣♦♦② ♥t♦r ♠♥s tt s t♦ ♦♥r s ♦♦r♥t♦r r t ♦♦r♥t♦r ♥♦ s ♥ ♣ ♦♥t ♠sr♠♥t tr♦②

♠♣s s♣♦♥s ❯tr ❲ ♥ ❯❲ ♦s

❯❲ ♥♦s ♥r② s ♦r s♥s♦r ♥t♦r ♣♦②♠♥t r st♦r ♦③t♦♥ ♥ tr♥ ♣♣t♦♥s ❬❪ r♦ ♦ t rt ttr♥s♠ss♦♥ t② ♦ r♦♥ tr♣ ② ♠sr♠♥ts t♥ ♣rt♦♣r s s r s♦ ♦r♥③ ♥ ♥tr③ t♦♣♦♦② ♥t♦r ♥ t♦♦r♥t♦r s ♥ ♣ ♦♥ t ♠sr♠♥t tr♦② ♠♥ s♣t♦♥s ♦ t❯❲ s r ❬❪

rq♥② GHz ♥ ♥t MHz ♦t♦♥ r♥t ♥r② Ps t ②♥ P ① s♥stt② dBm t♥ GHz ♥ GHz

♠♦♥ ♦tr t ♠♣s s♣♦♥s ❯tr ❲ ♥ ❯❲ ♥♦s ♣r♦ r♥♥t♠ ♦srs ♥ rt② ♦♥rt t♦ r♥ ♦srs ♥ t♦♥t♦ t♦s ♦srs t s s♦ ♣r♦ t ♥♥ ♥r② ♣r♦ t t♠rs♦t♦♥ ♦ 1 ns


sr♣t♦♥ ♦ t t ♥r♦s

P st ❲ ♠sr♠♥t ♠♣♥ s r② r ♥ s ♠ ♦ r♥t s♥r♦s ♥♥ sr s s♥r♦s ♦♥ ♦ t♦s s s♥r♦sts ♥t♦ ♦♥t r ♥♠r ♦ stt♦♥s ♥♦♥ ♠♦ ♥♦s r♥t s♥ r♥t st② stt♦♥s ♥ t ♦♦♥ ♦♥② s♣ s♥r♦s ♥st

s♥r♦ trs t r♦♠ s r ♥♦s r stt ①♣t t♠♦♥ tr♦② s s♥r♦ s s ♦r ♠♦ ♣r♠trs ①trt♦♥

s♥r♦ trs t ♦♥ ♥ r♦♥ trt ♣♦♥t r ♣♦♥t r②m ♦ ♣r♥ tr ♦r t ♠t tr♦② s s ♠♦ ♥♦sr stt

♥r♦ ♦ ♣r♠trs ①trt♦♥

♥ s♥r♦ tr♦② t ❩ ♥♦ ♦♦r♥t♦r ♥ ♥ ❯❲ ♥♦ ♦♦r♥t♦r s ♥ ♠♦ ♦♥ r♥t ♠sr♠♥t ♣♦♥ts ♥ t P ♥♦s ♣♦♥ts r r ② t ♦t ♦♥ r ♥s t ♥ r ♣ ❯❲ s ♣r♦ r♥s ♦srs t♥ ♦tr ♥ t♥t tr♦② ♥ s♦ ❩ s ♣r♦ ♦srs t♥ ♦tr♥♥ t ♥♦ ♣ ♦♥ t tr♦②

♥r♦ st♥ s♥r♦

s♥r♦ s t st s♥r♦ ♦r t ♣t♦r♠ ♥ ♦rt♠ t♦♥s♥ ts s♥r♦ t s ①♣t t♦s ♠ ♦♥ t tr♦② r stt ❯❲ s ♥ t ❩ s st♥ t t s♠ ♣♦st♦♥ s ♥ s♥r♦ r s♦s t ♣♦st♦♥s ♦ t ♦ ♥♦r ♥ t ♥♦r r♣rs♥t② r ♦ts ♥ r♥ tr♥s rs♣t② r♥t tr♦② ♣♦st♦♥s ♠sr♠♥t ♣♦♥ts r r♣rs♥t ② t ♦ts r s♣s t ♦ t♠sr♠♥ts ♣♦♥ts rr♦ ♦rrs♣♦♥s t♦ ♦♥t♦s st ♦ ♠sr♠♥ts ♥♠r ss♦t t♦ t t ♥ t ♦rrs♣♦♥s rs♣t② t♦ t rst♥ t st ♠sr♠♥t ♥① ♦ t st

r♥ tt ♦s r♦♠ ♥r♦

♥ ♦rr t♦ t♦ t ♦③t♦♥ ♦rt♠s t ♣r♠trs ♦ t ♣t♦ss♠♦ ♦r t ♥♦s ♥ t rr♦r st♥ ♠♦ ♦r t ♦ ♥♦s t♦ st♠t ♦ ♣r♠trs r ①trt r♦♠ s♥r♦

r♥ ❩ Pt♦ss ♦♥ ♦

rs ♥ r♣rs♥t rs♣t② t r ♣♦r ♥ t r♦♥trt st♥s s ♥t♦♥ ♦ t ♠sr ♣♦♥ts ♦r s♣ ♠sr♠♥t ♣♦♥tsr ♣♦r ♠sr♠♥ts r ♣r♦r♠ r ♦ts r♣rs♥t t ♠♥ ♦t ♠sr♠♥t t ♥ ♦sr tt t r ♣♦r s ② ♦rrt t♦ t

sr♣t♦♥ ♦ t sr♠♥t ♠♣♥

0

4

8

12

14

18

22

26

30

42

64

97

103

107

111

115

117

121

15

40

41

42

43

44

45

52

53

54

55

58

60

61

62

30 38

39

46

47

48

49

Measured pos itions

UWB/TOA Anchors

UWB/RSS Anchors

♥r♦ sr♠♥t ♣♦♥ts ♥ ♥♦r ♣♦st♦♥s s♣r♠♣♦s ♦♥ t♥ ♠♣ ♦ t ♠sr♠♥t ♠♣♥ r ♦ts r♣rs♥t t ♣♦st♦♥ ♦t ❯❲ ♦ s t r♥ tr♥ r♣rs♥ts t ♣♦st♦♥ ♦ t ❩ s ♦ts ♥ tr ss♦t ♥♠r r♣rs♥t ♠sr♠♥t ♣♦♥t


40

41

42

43

44

45

15

52

53

54

55

58

60

61

62

38

39

46

47

48

49

30

Measured positions

UWB/TOA Anchors

UWB/RSS Anchors

sr♠♥t ♥ ♥♦rs ♣♦st♦♥s

0

398094

4060

95

118

119126

132127

140140

147

139133

61 79

♥①s ♦ t ♠sr ♣♦st♦♥s

♥r♦ ♦t rs r♣rs♥t t ♥r♦ sr♣t♦♥ ♥ s r♣rs♥t t r♥t tr♦② ♣♦st♦♥s t ♦ts ♥ t ♣♦st♦♥s ♥ ♥♠ ♦♦ ♥ ♥♦r ♥♦s t r ♦ts ♥ r♥ tr♥s rs♣t② ♥ s s♣r♠♣♦s t ♠sr♠♥t ♥①s ♦ t tr♦② trt♦r② ♥♠r ss♦tt♦ t t ♥ t ♦rrs♣♦♥s rs♣t② t♦ t rst ♥ t st ♠sr♠♥t♥① ♦ t st

sr♣t♦♥ ♦ t sr♠♥t ♠♣♥

st♥ s ①♣t ♥ s ♦ str♦♥ r ♣♦r t rt♦♥ t♦ s♠ st♥s r r♥ t ♦t t t ♣♣rs tt tr s s♥♥t ♥♠r ♦r♦ ♥ rtr♥ t strt ♠♥♠ ♦ −81dBm t② tt ♦rrs♣♦♥s t♦ t s♥stt② ♦ t rr s s♦ s r② trs♣t t♦ ts rt♦♥ t♦ st♥ ♥ t ♦♦♥ t② ♥t ♥ ♦♥sr ♦r ttr♠♥t♦♥ ♦ t ♣t ♦ss ♠♦ ♣r♠trs

❯s♥ t ♦♥♦r♠ s♦♥ ♠♦ qt♦♥ t ♣t ♦ss ♠♦ ♣r♠trs ♦r ♦t ♦♥♥ ♥ ♠t♥ ss r ①trt ② ♥♦t♥ RSS t r RSS0 t r t d0 d t st♥ ♥ ss♠♥ Xσ s ③r♦♠♥ ss♥ r♥♦♠ r r♣rs♥t♥ t s♦♥ t ♥ rtt♥

RSS = RSS0 − 10np log10(d

d0) +Xσ

② ♣♣②♥ ♥r rrss♦♥ ♦♥ t ♠sr t t ♠♦ ♣r♠trs ♥ ①trt s ♦ t ♠♦ ♣r♠trs r r♣♦rt ♥ ♥ t rst♦ t rrss♦♥ s ♣r♦ ♥ r

❩ ♣t ♦ss ♠♦s ①trt ♦r s♥r♦

❩ ♦ RSS0 dB np σ dB

♥ t♦s ①trt s t s ♥♦t tt t r ♣♦r s r② r rt② s rt② s s♦ ♥ ♦sr ♦♥ t stt ♥s s ♥①♠♣ t r ♣♦r s ♥t♦♥ ♦ t ♠sr♠♥t ♣♦♥t ♦ stt ♥s s♥ ♥ r ♥ tt stt stt♦♥ t ♣♦r s ♥ r② ♥ r♥ ♦ ♠♦st dB ♥tr♣rtt♦♥ ♦ t♦s rt♦♥s s ♦r ♠♥ st♥t♦♥ t♥stt ♥ ②♥♠ s♦♥ r ①sts ♥ t ♣♣r r♦♥ st tt♥t♦♥r♥t rt② stt s ♣r♦② ♦rrs♣♦♥ t♦ rt♥ ♦♠♥t♦♥ ♦srt ♦♥rt♦♥s ♦ t ♥ s ♦♣♥ ♦r ♦s ♦♦rs ♥ t ♦r r♦♥ t♦sr tt♦♥s r ♣r♦② t♦ ♦tr str♦♥ ②♥♠ s♦♥ t♦ ♠♥♦s ♠♦♥ ♥t♦ t ♥♥ s ♦♥sq♥ t ①sts s♦♠ sss ♦t ①♣♦t♥♦r s♠♣ ♣t ♦ss ♠♦ ♥ s ♦♠♣① ♥ r stt♦♥s s ♣♦♥t st♦ ♣t ♥ ♠♥ ♥ t rtr ♠t♦ t♦♥ ♦r ♥r② t st s ♦t r ♣♦r ♥♦r♠t♦♥ s qt ♦♠♣① ♣r♦♠ ♥ ♥ ♥ ♥r♣r♥t♥♣♣r♦s t♦ ts r ♣r♦♣rt② ♦ ♥ r② s♥st t♦ ♥♥ ♠♦t♦♥s♥ ♥② s ♥ tr t s t♥q ♥ t♦♦ ♦r ♥rst♥♥ ♥♣rt♥ t ss♦t♦♥ ♦ r ♣♦r ♥ st♥ ♦r r ♣♦r ♥ ♣♦st♦♥s ♥ sst


0 20 40 60 80 100 120 140Measure number

85

80

75

70

65

60

55

50

45

rss

dB

m

30- 18

0 20 40 60 80 100 120 140Measure number

85

80

75

70

65

60

55

50

45

40

rss

dB

m

46- 18

0 20 40 60 80 100 120 140Measure number

82

80

78

76

74

72

70

68

66

64

rss

dB

m

38- 18

0 20 40 60 80 100 120 140Measure number

85

80

75

70

65

60

55

rss

dB

m

49- 18

0 20 40 60 80 100 120 140Measure number

85

80

75

70

65

60

55

50

45

40

rss

dB

m

39- 18

0 20 40 60 80 100 120 140Measure number

85

80

75

70

65

60

55

50

rss

dB

m

48- 18

♣♦r s ♠sr♠♥t

0 20 40 60 80 100 120 140Measure number

0

5

10

15

20

25

Dis

tance

[m

]

30- 18

0 20 40 60 80 100 120 140Measure number

0

2

4

6

8

10

12

14

Dis

tance

[m

]

46- 18

0 20 40 60 80 100 120 140Measure number

6

8

10

12

14

16

18

20

22

Dis

tance

[m

]

38- 18

0 20 40 60 80 100 120 140Measure number

2

4

6

8

10

12

14

16

Dis

tance

[m

]

49- 18

0 20 40 60 80 100 120 140Measure number

0

2

4

6

8

10

12

14

16

18

Dis

tance

[m

]

39- 18

0 20 40 60 80 100 120 140Measure number

2

4

6

8

10

12

14

16

18

20

Dis

tance

[m

]

48- 18

st♥ s ♠sr♠♥t

♥r♦ ♠sr ❩ r ♣♦r s ♥t♦♥ ♦ t♠sr ♣♦♥ts r♦♥ trt st♥ s ♥t♦♥ ♦ t ♠sr ♣♦♥t r ♦ts ♦♥ t r ♣♦r ♦rrs♣♦♥ t♦ t ♠♥ s ♦ r ♣♦r ♦r t♥ ♠sr ♣♦st♦♥s

sr♣t♦♥ ♦ t sr♠♥t ♠♣♥

101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

30

100 101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

46

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

38

101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

49

100 101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

global

101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

39

101

distance (m)

−90

−80

−70

−60

−50

−40

Pr

(dB

m)

48

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

♣♦r s ♥t♦♥ ♦ t st♥ ♦r t ♥♦rs r ♥ ♦rrs♣♦♥ t♦ t ①trt ♦♥ s♦♣ ♣t ♦ss ♠♦


0 100 200 300 400 500 600

Measure index

−85

−80

−75

−70

−65

−60

−55

−50

Pr

(dB

m)

49-38

0 100 200 300 400 500 600

Measure index

−85

−80

−75

−70

−65

−60

−55

−50

Pr

(dB

m)

49-39

①♠♣ ♦ ♣♦r ♠sr♠♥t t♥ stt ♥s t♥ ♥ ♥ t♥ ♥

r♥ ♥ ❯❲ ♥♥ rr♦r ♦

❯s♥ s♠r ♣r♦r ♦♥ t ❯❲ ♠t♥ t ♠sr st♥s t♦ ttr st♥s ♥ ss♠♥ ss♥ ♠♦ ♦r t rr♦r strt♦♥ t s ♣♦sst♦ r r♥♥ rr♦r ♠♦ ♦t♥ s ♦ t ♠♦ r ♣rs♥t ♥ t s ♥♦t tt t ♠♦ ♣r♠tr s ♥ ♦t♥ ② rt♥♦ t ♠sr t t♦t ♥② st♥t♦♥ t♥ ♥ stt♦♥s ♥rr t t t♦rt ♣r♦r♠♥s ♦ ❯❲ t♥♦♦② t s r②② tt ♥ stt♦♥s ♦t t ♠♥ r♥♥ rr♦r ♥ t st♥r t♦♥♦ s♥♥t② ♦r

❯❲ ♦ r♥♥ ♠♦ ①trt ♦r s♥r♦

♥r♦ ♠♥ r♥♥ rr♦r m st m

♦♠♣rs♦♥ t♥ ♠sr ♥ s♠t Ps

s ♣rt s t t♦ t t♦♥ ♦ t ♠t♦s ♣r♦♣♦s ♥ t tr rst♣trs ♦r tt ♣r♣♦s t ♠sr♠♥t ♥r♦♥♠♥t s ♥ rrt ♥ s②♥tt rs♦♥ ❲t t ♣ ♦ ts ♥r♦♥♠♥t t st s♣ ♠t ♣♣r♦s s t♦ ♣r♦ ♦ ♥ ♦srs ♦s s♠t s r ♦♠♣r t♦t ♠sr ♦♥ ♦r t ♠sr ♣♦st♦♥s s t ♣r♦♣♦s t♦♦ ♣r♦s ♦r s♣ tr♦② ♣♦st♦♥s ♥ r ♦♠♣r t♦ t ♠sr ♦♥s ♥t ♠sr ♦ ♥ ♦srs r s t♦ t P ♣♦st♦♥♥♣r♦r♠♥s r t ♥ ♦♠♣r t♦ ♣♣r♦ ♥② t t st ss t♦ t t ②♣♦tss tst♥ ♠t♦ ♥st P ♥ ♣♣r♦

♦♠♣rs♦♥ t♥ ♠sr ♥ s♠t Ps

s ♥ r♥ t♥ ♠srs ♥ s♠t♦♥s ♦r t ♥ r♥s ♦srs

♥♦r t②♣ P rr♦r ❱r♥

❩ dBm dBm❯❲ r♥s m m

t♦♥ ♦ t t ♣♣r♦ ♦♥ sr t

s ♣rt ♦♠♣rs s♠t P ♦t♥ t t ♠t ♣♣r♦ sr♥ t t♦s ♠sr r♥ t ❲ ♠sr♠♥t ♠♣♥ ❬❪

♦♠♣rs♦♥ t♥ ♠srs ♥ s♠t♦♥s ♥ ♣r♦r♠ ♦r t ♥♦rs ♦♥ t tr♦② trt♦r② rs ♥ s♦ t♦s ♦♠♣rs♦♥st♥ ♠sr♠♥t ♥ ♠t s♠t♦♥ ♦r r ♣♦rs ♥ r♥s rs♣t② rr♦r s ♥ s t r♥ t♥ ♠srs ♥ s♠t♦♥s s♥ t s r♣rs♥t ② t r r♦sss ♠sr♠♥ts ♣♦♥t ♥①s r rt t♦t ♥①s ♣rs♥t ♥ r

t s t♦ ♥♦t tt t ♠trs ♣r♠trs ♦ t r ♠sr♠♥t ♥r♦♥♠♥t r ♥♦t ♥♦♥ ♥ ts ♥ ♠♦ t st♥r ♣r①st♥ ♠trsr♦♠ t s②♥tt ♣t♦r♠ ts ①♣t ♦r s♦♠ s♣ ♥ ♥ ♣♦st♦♥s rt rr♦r ♥ r② ♣r♦② t♦ tt ♦ ♥♦ ♦ t ♥r♦♥♠♥tt s ♥♦t tt t st s♣ ♠t ♣♣r♦ ♣r♦s ♦♦ st♠t♦♥ ♦♦t t r ♣♦rs ♥ t r♥ ♦srs t ♦r s s♦ ♦♥r♠② t s ♥ rr♦r r♥s ♣rs♥t ♥ t

t♦♥ ♦ t ② r♥ ♣♣r♦ ♦♥ sr t

♥ ts ♣rt t ♠sr ❯❲ r ♦♠♣r t♦ t s♠t ♦♥ ♦r t♦♣♦st♦♥s ♦ t tr♦② t♦ r♥t ♦♥rt♦♥s ♦ tr♦②♣r ♥♦ ♣♦st♦♥r♣rs♥t ♥ r r ♦♥sr rst ♦♥rt♦♥ ♦rrs♣♦♥s t♦ stt♦♥ r t tr♦② s ♣♦st♦♥ ♥ ♦♦r ♥ t ♣r ♥♦ s ♥①t t♦ ♦ r♦♦♠ s♦♥ ♦♥rt♦♥ ♦rrs♣♦♥s t♦ stt♦♥ r ttr♦② ♥ t ♣r ♥♦ r ♦t ♣♦st♦♥ ♥t♦ ♦rr♦r

r♦♠ t♦s ♦♥rt♦♥s t ♦♠♣tt♦♥ st♣s sr ♥ r ♣♣ ♥♦rr t♦ ♦t♥ r②s r t♥ r②s r

♥ rs t ♥rt s ♦♠♣r t♦ t ♠srs ♦t♥ r♦♠ tt ♦t t♦ t ❯❲ ♠sr t r ♥♥ ♥r② ♣r♦ t t♠ rs♦t♦♥ ♦ 1ns st♦r ♥t♦ rr r s ♦♥sq♥ t st♦r ♥r②♣r♦s ♥ r♥ s♥ t ♠sr r♥♥ st♠t♦♥ t♦ ♦♠♣rt t s♠t♦♥ ♥ t♦♥ ♠sr s♠♣ s 16bit ♥tr ♣r♦♣♦rt♦♥ t♦ t ♦ r ♣♦r ♦r t ♦♠♣rs♦♥ ts ♥tr s ♥♦♥rt ♥t♦ tr rs♣t dBm s ♦t tt ts ♦♥rs♦♥ s ♥ ♣♣r♦①♠t♦♥ ♦ t t r s t♦ t ♦ ♥♦r♠t♦♥ ♦♥ ♦t t ♠ss♦♥♣♦r ♥ t ♠①♠♠ r ♣♦r


0 50 100 150 200

Measurents points index

85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)

Received Power from anchor nb 30Estimated

Measured

Error

0 50 100 150 200


85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)


Measured

Error

0 50 100 150 200


85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)


Measured

Error

0 50 100 150 200


85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)


Measured

Error

0 50 100 150 200


85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)


Measured

Error

0 50 100 150 200


85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)


Measured

Error

0 50 100 150 200


85

80

75

70

65

60

55

50

45

40

Pow

er

(dB

m)


Measured

Error

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

0

5

10

15

20

25

30

35

40

Err

or

(dB

m)

♦♠♣rs♦♥ s♠t♦♥ ♠srs ♦ t r ♣♦r ♦r ❩ ♥♦r♥♦s s ♥t♦♥ ♦ r♥t ♠sr♠♥t ♣♦st♦♥s ②♥ ♦ts r t ♠srt ♦ts r s♠t t r r♦sss r t rr♦r ♦r trt♦r② ♣♦♥t

♦♠♣rs♦♥ t♥ ♠sr ♥ s♠t Ps

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)

Range from node nb 15Estimated range

Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30R

ange (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0 20 40 60 80 100 120


0

5

10

15

20

25

30

Range (

m)


Measured range

Error

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Err

or

(m)

♦♠♣rs♦♥ s♠t♦♥ ♠srs ♦ t r♥s ♦r ❯❲ ♥♦rs ♥♦ss ♥t♦♥ ♦ r♥t ♠sr♠♥t ♣♦st♦♥ ②♥ ♦ts r t ♠sr t ♦ts r s♠t t r r♦sss r t rr♦r ♦r trt♦r② ♣♦♥t


10 15 20 25 30

5

10

15

20

25

trolley

peer node

♦♥ ♥♦ ♣♦st♦♥s10 15 20 25 30

5

10

15

20

25

trolley

peer node

♦♥ ♥♦ ♣♦st♦♥s

♦ ♣♦st♦♥ ♦♥rt♦♥s ♦ t r♦② ♥ t ♣r ♥♦ s ♦r ♦♠♣tt♦♥ ♥ ♦♥rt♦♥ s stt♦♥ ♥ ♦♥rt♦♥ s stt♦♥

♦♥ r②s ♦♥ r②s

r②s ♦t♥ r♦♠ t ♦♠♣tt♦♥ ♦r ♦♥rt♦♥ ♥ ♦♥rt♦♥

♦♠♣rs♦♥ t♥ ♠sr ♥ s♠t Ps

♦♥ r②s ♦♥ r②s

r②s ♦t♥ r♦♠ t ♦♠♣tt♦♥

0 10 20 30 40 50 60 70 80

de lay (ns)

0.0015

0.0010

0.0005

0.0000

0.0005

0.0010

0.0015

linearlevel(V)

Simulation

0 10 20 30 40 50 60 70 80

de lay (ns)

0.0

0.2

0.4

0.6

0.8

1.0

Normalizedlinearlevel

UWB CEA Measure

♦♥ s ♥♥ ♥r② ♣r♦

0 10 20 30 40 50 60 70 80

de lay (ns)

0.003

0.002

0.001

0.000

0.001

0.002

0.003

linearlevel(V)

Simulation

0 10 20 30 40 50 60 70 80

de lay (ns)

0.0

0.2

0.4

0.6

0.8

1.0


UWB CEA Measure

♦♥ s ♥♥ ♥r② ♣r♦

♥♥ ♠♣s rs♣♦♥ss ♥rt s♥ t ♣r♦♣♦s r② tr♥ t♦♦♦♠♣r t♦ t ♥♦r♠③ ♠sr ♥♥ ♥r② ♣r♦ ♥ t r♦♥ trt② s 32.3 ns rs t ② ♦rrs♣♦♥♥ t♦ t ♠sr t♦ ② r♥♥ s32.8 ns ♥ t r♦♥ trt ② s 21.8 ns rs t ② ♦rrs♣♦♥♥ t♦t ♠sr t♦ ② r♥♥ s 22.17 ns r ♥ s♣rt♥ t r② ♣rt r♦♠t t ♣rt r♣rs♥t t r♦♥ trt ②


♦r t s ♦ t ♦♠♣rs♦♥ t ♦ ♦♥♥♥t t♦ ♦sr t ♦rrs♣♦♥♥♥♥ ♥r② ♣r♦ ♦ t s♠t t ♦r♥ t♦ ❬❪ t ♠♣s rs♣♦♥sh ♦ ♣r♦♣t♦♥ ♥♥ ♥ rtt♥

h(τ) =∑

i

aiδ(τ − τi).

r ai ♥ τi r t tt♥t♦♥s ♥ tr ss♦t ♣r♦♣t♦♥ ② rs♣t②♥ t s ♣♦ss t♦ ♣r♦ t♦ t srt③t♦♥ ♦ t ♥♥ ♠♣s rs♣♦♥s s♥t ♦♦♥ ♦r♠

hl =∑

i

ais♥(l − τiW ).

r l ♥W r t♣ ♥① ♥ t ♥ ♦ t s②st♠ rs♣t② lt t♣ ♥ ♥tr♣rt s t s♠♣ l/W t ♦ t ♦♣ss tr s♥ ♥♥ rs♣♦♥shb(τ) ♦♥♦ t s♥(Wτ)

sr♥ tt h s t ①♣rss♦♥ ♦ t tr♥s♠ss♦♥ ♥♥ ♦r ♥ rq♥②♣r♦♣♦s ♥ t s ♣♦ss t♦ srt③ t s♠t ♣r♦♣t♦♥ ♥♥ t♦♦t♥ t ss♦t ♥♥ ♥r② ♣r♦

0 10 20 30 40 50 60 70 80

delay (ns)

0.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

ed lin

ear

level

Simulation

0 10 20 30 40 50 60 70 80

delay (ns)

0.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

ed lin

ear

level

UWB CEA Measure

♦♥ ♥♥ ♥r② Pr♦ ♦♠♣rs♦♥

0 10 20 30 40 50 60 70 80

delay (ns)

0.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

ed lin

ear

level

Simulation

0 10 20 30 40 50 60 70 80

delay (ns)

0.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

ed lin

ear

level

UWB CEA Measure

♦♥ ♥♥ ♥r② Pr♦ ♦♠♣rs♦♥

♥♥ ♥r② ♣r♦ ♦ s♠t♦♥s ♥ ♠sr ♥ t r♦♥ trt② s 32.3 ns rs t ② ♦rrs♣♦♥♥ t♦ t ♠sr t♦ ② r♥♥ s32.8 ns ♥ t r♦♥ trt ② s 21.8 ns rs t ② ♦rrs♣♦♥♥ t♦t ♠sr t♦ ② r♥♥ s 22.17 ns r ♥ s♣rt♥ t r② ♣rt r♦♠t t ♣rt r♣rs♥t t r♦♥ trt ②

s♠t ♥ ♠sr ♥♥ ♥r② ♣r♦ ♦ ♦t stt♦♥s r ♦♠♣r♥ rs ♥ t ♥♦r♠③ ♣♦r ♥♦r♠③ ♣♦r r♣rs♥tt♦♥ s ♥ ♦s♥ ♦t t♦ t ♥rt♥t② ♦♥ ♥trdBm ♦♥rs♦♥

t♦♥ ♦ ♣r♦♣♦s ♦③t♦♥ ♦rt♠s

♥ t♦ t ♠ss♥ ♠ss♦♥ ♣♦r ♦ t♦s ♥♦r♠t♦♥ ♣r♥ts t♦♦t♥ t ♦rrt r ♣♦r ♥ s♠t♦♥ t ♥r② s s♦ ♥♦r ② t♥♥♦♥ ♦ t tr♥s♠ttr ♥ rr ♥t♥♥ ♦r♥tt♦♥s ♥ rt♦♥ ♣ttr♥s♦r s ①♣♥ ♣r♦s② t ♠ r s ♥♦t s♦ ♠ t♦ ♣r♦ strt②t s♠t♦♥ ♦ t ♥♥ t t♦ rtr t ♠♥ ♦♥trt♦rs ♦ t ♥ t ♣♦st♦♥ ♦ t strs ♥ t s♥s ♦ t ❱♥③ ♠♦ ❬❪ ♦trqr ♦r ♦③t♦♥ ♣r♣♦s

♦♥sr♥ r ♦rrs♣♦♥♥ t♦ stt♦♥ t ♥ ♦srtt t rst ♣t s t♦ st♠t t ② t 32.8ns s ♦rrt② rtr ♥ tr♠s ♦② ♥ s♠t♦♥ s s♦♥ str♦♥ ♦♥trt♦♥ t ♦t 45 ♥s s ♦sr♦♥ ♦t s♠t♦♥ ♥ ♠sr t 35 ♥s str♦♥ ♦♥trt♦♥ s ♦sr ♦♥ t s♠t♦♥ ♠ss♥ ♦♥ t ♠sr♠♥t t s ♣r♦② s ② ♥ ♦st ♥♦♥sr ♥ t s②♥tt ♥r♦♥♠♥t ♦t♥ tt ♦♥trt♦♥ ♦r ♥st♥ ♥♦r♥trs ♥ t♥ ♥t♦ ♦♥t ♥ t s♠t♦♥ rs t r♦♦♠ ♥♦♥ tt ♦♥rt♦♥ s ♦♣♥ s♣ t sr s ♥ ♦♠♣trs ❬❪ ♦♥sr♥r ♦rrs♣♦♥♥ t♦ stt♦♥ ♦♥trt♦♥ ♦rrs♣♦♥♥ t♦ trt ♣t s ♦sr ♦t ♥ s♠t♦♥ ♥ ♠sr♠♥t t 21.8 ♥s ♦ tt ♦♥trt♦♥ s r ♥ s♠t♦♥ t♥ ♥ ♠sr♠♥t ♦♥trr② r♦♥30 ♥s s♦♥ ♦♥trt♦♥ s ♦sr t r ♥ ♠sr♠♥t t♥ ♥s♠t♦♥ ♠♦st ♣r♦ ①♣♥t♦♥ ♦ ts r♥ s t ♥t♥♥ ♦r♥tt♦♥♥ s♠t♦♥ t ♠♥ ♦s ♦ ♥t♥♥ ♥ ♦tr ①♣♥♥ t str♦♥♦♥trt♦♥ t t ①t r♦♥ trt ② ♥ ♠sr♠♥t t s ② tt t♠♥ ♦s ♦ ♦t ♥t♥♥s r ♥♦t ♥ ♦tr ①♣♥♥ tt t rt ♣t♦♥trt♦♥ s ♦r t♥ t tr rr ♦♥

t♦♥ ♦ ♣r♦♣♦s ♦③t♦♥ ♦rt♠s

s st♦♥ ♦ss ♦♥ t t♦♥ ♦ t ♦③t♦♥ ♦rt♠s ♦♥ t rt st rst P s t ♦r ♥♦♥②r ♥ ②r ss ♦♥ t ♦ tst ♦♥ t ②♣♦tss tst♥ ♠t♦ s t ② rstrt♥ t ♥♠r ♦♦srs ♦ ♥ t♦ ♦♠♣♥t t t ♠t♦

t♦♥ ♦ t Pr♦r♠♥s ♦ P

♥ ts st♦♥ t P ♦③t♦♥ ♦rt♠ s ♣♣ ♦♥ t ♠sr t ♥♦♠♣r t♦ ♥ tr♦♥♦s ♣♣r♦①♠t♦♥ ♦rt♠ ♥t③ t r♥♦♠ ❬ ❪ ♦r tt ♣r♣♦s t ♠♦s ①trt ♥ sst♦♥ ♥ ttst r♦♠ s♥r♦ r ♦♥sr

♦t ♥ P ♦rt♠s r ♦♠♣r t t ♣ ♦ ♠t ♥st②♥t♦♥s s ♦ ♣♦st♦♥♥ rr♦r s♥ r♥t ♦♥rt♦♥s

❩ r ❯❲ ♦ r ❩ ♥ ❯❲ ♦ r


♦r ♦ ts ♦♥rt♦♥s t ♥♠r ♦ ♦sr ♣♥s ♦♥ t ♣♦st♦♥♦ t tr♦② ♦r ♦sr ♦ ♥ P t ♥ ♣♦st♦♥ ss t♦ ♣r♦r♠ t ♣♦st♦♥ st♠t♦♥ r s♦s t ♣r♥t ♦ ♠srs♥♦♥ ♥ ♥♠r ♦ P ♦r ♥st♥ ♦ t st♠t ♣♦st♦♥ ♥ st♠t s♥ ♦♥② ♦s

2 3 4 5 6

number of used ToA

0

5

10

15

20

25

30

Percentageofpositionestimated

usingXnumber

ofToAs

7

Pr♥t ♦ t ♥♠r ♦ ♦ s

4 5 6 7

number of used RSS

0

10

20

30

40

50

60

70

Percentageofpositionestimated

usingXnumber

ofRSSs

Pr♥t ♦ t ♥♠r ♦ s

2 3 4 5 6 7 8 9 10 11 12 13 14

number of used ToA and RSS

0

5

10

15

20

25

30

Per

cen

tag

eo

fp

osi

tio

nes

tim

ated

usi

ng

Xn

um

ber

of

To

As

and

RS

Ss

Pr♥t ♦ t ♥♠r ♦ ♦ ♥ s

st♦r♠s t♦ r♣rs♥t t ♥♠r ♦ Ps s ♣r t ♣♦st♦♥♦r t ♦♥rt♦♥ ♥♦♥ ♦♥② t ♦ ♦♥② t ♦r t ♦♥ ♦r ♥st♥ s♠t♦♥ ♥♦♥ ♦♥② Ps 15 ♦ st♠t ♣♦st♦♥ ♥ r③ t 5

r r♣rs♥ts t s ♦ ♣♦st♦♥♥ rr♦r ♦r ♦t P ♥ s♥♦♥② ♦ ♦srs t ♥ ♦sr tt P ♦t♣r♦r♠s t ♥ t r♠♦ ♦ rr♦rs s 90 ♦ t ♣♦st♦♥s r st♠t t ♥ rr♦r ss t♥ 3 ♠trss♥ P rs t s ♥ rr♦r ss t♥ 4 ♠trs ♥ tt stt♦♥♦r ♦r st♠t ♣♦st♦♥s t r rr♦r t ♣rs ♦♥ t P

♥ ♦♥trr② ♦♥sr♥ t ♦♠♣rs♦♥ ♦ s ♦ ♣♦st♦♥♥ rr♦r s♥ ♦♥② ♦srs ♦♥ r t ♦t♣r♦r♠s t P ♣♣r♦ s rst♥ ♣rt② ①♣♥ ② t ts ♦ P t♦ ♦♥ ♦♥str♥t ♥

t♦♥ ♦ ♣r♦♣♦s ♦③t♦♥ ♦rt♠s

s ①♣♥ ♥ st♦♥ t ♦♥str♥t s t t t ♣ ♦ ♠♦ t♠♦ ♦♥rts r ♣♦r ♥♦r♠t♦♥ ♥t♦ st♥ ♣s t♦ ♠t s♣r♦♥ ②♣② ♥ sr ♦♥str♥ts r s t ♦① rst♥ ♦ t ♠r♦ ♦♠tr ♦①s ♦♥s r r s ♦♥sq♥ t st♠t♦♥ ♦t ♣♦st♦♥ ♥ ss rt t♥ t s♦t♦♥

t tr♥ s s♦ ♦♥r♠ ♥ tr♦♥♦s s r r s ♥ t♦♥t♦ ♦ ♦sr s t ♥ ♦sr ♥ r t P ♦② ♦t♣r♦r♠s t ♦rt♠ t s♠s tt t ♦♥t r② t ♥t ♦ t ①tr ♥♦r♠t♦♥ t rst s ♦♥r♠ ② ♦♥sr♥ t s♠♠r③ ♣r♦r♠♥s ♥tr♠s ♦ ♦♦t ♥ qr rr♦r ♥ st♥r t♦♥ ♣rs♥t ♥ t ♥ ♦sr tt t P ♦rt♠ ♣r♦r♠s s t ♦r t♦t Ps ♥ tr♠s ♦ t ♦r ♥ ①♣♥ ② ts ♦ P t♦♦① t ♦♥str♥ts ♥ t ♦♥str♥t ♦① ♦ t ♦srs ♦♥t ♣t♦ rstrt t r r t ♣♦st♦♥ s st♠t

0 1 2 3 4 5 6 7 8 9


0.0

0.2

0.4

0.6

0.8

1.0

Cum

ula

tive

pro

bab

ilit

y

RGPA

ML

♦

0 5 10 15 20


0.0

0.2

0.4

0.6

0.8

1.0

Cum

ula

tive

pro

bab

ilit

y

RGPA

ML

0 1 2 3 4 5 6 7 8 9


0.0

0.2

0.4

0.6

0.8

1.0

Cum

ula

tive

pro

bab

ilit

y

RGPA

ML

♦

s ♦ ♣♦st♦♥♥ rr♦r ♦♠♣rs♦♥ t♥ ♥ P ♦rt♠ss♥ ❯❲ ♦ P ♥ ❩ P s♦♥ ♦ ♦ ♥ Ps

r s♦s t ♣♦st♦♥♥ rr♦r ♦ ♦t P ♥ ♦r ♣♦st♦♥


ttst ♣♦st♦♥♥ rr♦r ♦r s♥r♦

Ps ♠♥ ♣♦st♦♥♥ rr♦r ❬♠❪ st ♦ ♣♦st♦♥♥ rr♦r ❬♠❪♦rt♠s P P

❩ ❯❲ ♦

❩ ❯❲ ♦

♥①s ♦r t ♦♥sr s♥r♦s ♦t r ♥ r ♦rr♦♦rtt ♦srt♦♥ ♦ t P ♦② ♦t ♣r♦r♠s t ♠t♦ ①♣t ♦♥s♦♠ ♣♦♥ts t s ♥♦t tt ♦r t♦s ♣♦♥ts P ♦♠♠ts r rr♦rs ①♣♥♥t ♦r ♣r♦r♠♥s ♥ t r♠ ♦ r rr♦rs ♦ t rtr ♦r s♦ rqr t♦ ♦♠♣t② tr♠♥ t ♦r♥ ♦ tt ♣r♦♠ ♥ ②♣♦tss ♦ tt t♦s r rr♦rs ♦rrs♣♦♥ t♦ ♠t♠♦ stt♦♥ ♥ t♥ t ♣♦st♦♥st♠t♦♥ ♦ ♣r♦r♠ ♦♥ t r♦♥ ♣♦ss s♦t♦♥

♦③t♦♥ s ♦♥ ②♣♦tss tst♥

♥ ts st♦♥ t ♦③t♦♥ s ♦♥ t ②♣♦tss tst♥ ♠t♦ ♣rs♥t♥ s ♣♣ ♦♥ t t r♦♠ ♠sr♠♥t ♠♣♥ ♥ ♦ t ts♥♦♥tr ♥ ts ♠t♦ r② ♦♥ t qst♦♥ rt② ♦ t ♣t ♦ss s♦♥♠♦ s ♦r t ♥ t s ♥ s♥ tt t ♣♦r ♥♦r♠t♦♥ s st♦ t♥ ♠t ♠♦ s♦t♦♥s qt② ♦ ts s♦♥ s rt② rtt♦ t qt② ♦ t ♠♦ s ♦r t ♣♦r ♥♦r♠t♦♥ s ♠♥t♦♥ ♦r ♥ ♦t♥♥ t ♣r♠trs ♦ ♣t♦ss s♦♥ ♠♦ s ♥♦t ♥ s② ts t♦ r♥t ♥ ♦ ♦sr rt② ♥ t♦s ♦♥t♦♥s t ♠t♦ s♦♥ ②♣♦tss tst♥ s s♦♥ tr ♠ts ♥ s ♣r♦r♠ ♦rst tt t ssP ♥ ♥ ts st♦♥ t ♣r♠trs ♦ t ♣t ♦ss s♦♥ ♠♦ ♥ ①trt ♦♥ t s♠ s♥r♦ ♥st ♦ s♥r♦ ss♠♥ tt ♣♦♥t tt r ♥♦ r②♥ ♦♥ r② s♣ ♥ ts ♦r ♠♦ t ♦♥♣t ♦ t♠t♦ s ♣r♦♥ ♥ t ♦♦♥

r s♦s ♦♠♣rs♦♥ t♥ t ♠t♦ t P ♠t♦ ♥t P s t t ②♣♦tss tst♥ ♠t♦ P s♠t♦♥ s♥ r♥ ♦♥ t ♠sr♠♥t ♣♦♥ts ♦ s♥r♦ s♥ ♦ ♦srs t tst P ♥ ♦♦s♥ t ♦sr t t st ♦rt♠sr t t s♠ ♦srs ♣r♠trs rst ♦ t s ♥♦t tt ♦tP ♥ P ♣r ♦♥ ♦r ♣♦st♦♥♥ rr♦rs ♥r m➑♥ tt r♦♥t P ♠t♦ strts t♦ r♥tt t♦ t P r♦♠ m♦ ♣♦st♦♥♥rr♦r ♥ t s r② ② tt st♠t ♣♦st♦♥s ♦t♥ t ♣rs♦♥ ♦ mr ♦t♥ t t ♣ ♦ r② ♣rs ♦s ♥ stt♦♥s r ♥♦ ♠t ♠♦s♦t♦♥s ♣♣r ♦r ♣♦st♦♥♥ rr♦r ♦ m ♣r ♦r ♣r♦r♠s s ttP ♦r P ♦r P rs t ♠♦♥t ♦ r rr♦rs ♦♠♣rt♦ P t ♠tt♦♥ ♦ r rr♦r s s♦ ♦sr ♦♥ r ♦♠♣r♥t ♣♦st♦♥♥ rr♦rs ♦ P ♥ P ♦r ♠sr♠♥t ♣♦♥ts ss

t♦♥ ♦ ♣r♦♣♦s ♦③t♦♥ ♦rt♠s

0 20 40 60 80 100 120 140

measurement point index

0

1

2

3

4

5

6

7

RM

SE

[m]

RGPA

ML

♦

0 20 40 60 80 100 120 140


0

5

10

15

20

RM

SE

[m]

RGPA

ML

0 20 40 60 80 100 120 140


0

2

4

6

8

10

RM

SE

[m]

RGPA

ML

♦

rr♦r ♦ ♣♦st♦♥♥ s♥ P ♦r s ♥t♦♥ ♦ t ♠sr♠♥t♣♦♥ts ♦r ❯❲ ♦ P ♥ ❩ P s♦♥ ♦ ♦ ♥ Ps


r t ♣♦st♦♥♥ rr♦rs ♦ P r r t♥ t♦s ♦ P ♦rrs♣♦♥t♦ s s♦♥ ♦♥ t ♦ r s t ♥ ♦sr ♥ t t P ♥ts ♦ t ♦ r② ♣r♦♥ t ♦r ♠♥ rr♦r ♦ ♣♦st♦♥♥ ♦♠♣rt♦ ♦t P ♥ t s ♥♦t tt P s r st♥r t♦♥t♥ t ♣♦st♦♥♥ rr♦r t♥ s ♥ ①♣♥ ② t ♠t♦ ts ♥rr♦r s♥ P ♦rrs♣♦♥s t♦ s tt♦♥ s q♥t t♦ st tr♦♥ r♦♥ t r♦♥ r♦♥ ♥ st♥t r♦♠ t r♦♥ trt ♣♦st♦♥s ♥ts s rt rr♦r

0 2 4 6 8 10


0.0

0.2

0.4

0.6

0.8

1.0

Cu

mu

lati

ve

pro

bab

ilit

y

RGPA HT

RGPA

ML

♦♠♣rs♦♥ ♦ ♣♦st♦♥ st♠t♦♥ t♥ P ♥ P t t②♣♦tss tst♥ ♦r ♦s ♥ ♦t t ♥ P s t s♠ ♦srs t♦ ♣r♦r♠ t ♣♦st♦♥ st♠t♦♥ P t t ②♣♦tss tst♥ sst ♦sr t♦ ♦♦s t♥ t ♠t ♠♦ s♦t♦♥s tt ♠t ♣♣r

0 20 40 60 80 100 120 140


0

1

2

3

4

5

6

7

8

RM

SE

[m]

RGPA HT

RGPA

rr♦r ♦ ♣♦st♦♥♥ s♥ P ♦r P t t ②♣♦tss tst♥ s ♥t♦♥ ♦ t ♠sr♠♥t ♣♦♥t

♦♥s♦♥

ttst ♣♦st♦♥♥ rr♦r P s P

t♦ ❬♠❪ st ♦ ♣♦st♦♥♥ rr♦r ❬♠❪

P

P

♦♥s♦♥

s ♣tr s ♦s ♦♥ t ①♣♦tt♦♥ ♦ t P ❲ ♣r♦t ♠sr♠♥t ♠♣♥ rst ♥ rt sr♣t♦♥ ♦ t ♠sr♠♥t ♠♣♥ s ♥♣r♦ ♥♥ t s s t♥♦♦s ♥ sr♣t♦♥ ♦ t sts♥r♦s s♣ st♦♥ s ♥ t t♦ t r♦ ♥♥ ♣r♠trs ①trt♦♥ r♦♠ ♦♥ ♦ t♦s s♥r♦s ♥ t t♦♥ ♦ t ♠t ♣♣r♦ s♥ ♣r♦r♠ ♦♥ t ♦ t st ♥ ♦♠♣rs♦♥ s♠t♦♥ ♠sr♠♥ts ♦t ♦ ♥ s ♥ ♣r♦ ♠t♦♥ s♦♥ r② ♣r♦♠s♥ rsts ♥rr ♦ t ♥♦♠♣t ♥♦ ♦ t ♦♥sttt ♠trs s ♥ t ♥r♦♥♠♥t ♦t♥ r♦♠ t t♦♦ s♦ ♥ ♦♠♣r t♦ t♦s ♦t♥② ♠sr s♣② t rsts s♦♥ tt t s t♦ ♣rt qtrt② t ♥♥ ♦ ♠t♣t ♦♥trt♦rs ♥ t P s ♥ tst♦♥ t ♠sr t st t♦ t t r② ♥ tr♠ ♦ ♣♦st♦♥ st♠t♦♥ ♥ss ♦ ♦srs P ♣♣r♦ s ♣r♦ rsts tt ss rt t♥ s♦t♦♥ s rst s ①♣t t♦ t t t② ♦ P t♦ ♠♥ ♦srs ♥ t ♦tr ss P s ♦t♣r♦r♠ t s s♦t♦♥♥② t ②♣♦tss ♠t♦ s ♥ ♣♣ ♦♥ stt♦♥ r ♦ ♦sr♥ s♥ s ♥ tt stt♦♥ t ♠t♦ s s♦♥ ts ♥trst♥st ♣♣r♦ ② r♥ t ♣♦st♦♥♥

♣tr

♦♥s♦♥

♠ ♦ ts tss s t♦ rss t ♣r♦♠ ♦ t ②♥♠ s♠t♦♥ ♦ tr♦♠♥t ♣r♦♣t♦♥ ♥ ♥♦♦r ♥r♦♥♠♥ts ♦r ♦③t♦♥ ♣r♣♦s ♦ss ♥ ♠♥② ♠ ♦♥ tr s♣ts ♦ t s♠t♦♥ t r♦ ♣r♦♣t♦♥t ♦③t♦♥ ♦rt♠s ♥ t s♥ ♦ t ②♥♠ ♣t♦r♠ ♦t♥ rsts♥ t r ♦♥s♦♥s r s♠♠r③ ♥ tr rsr s r ♣r♦♣♦s

♦♥s♦♥s

♦♥♣t ♦ s♥tr s ♥ ♥tr♦ t♦ sr t ♦sr ♣t♣rsst♥② t ♥ s ♦ r♥t ♠♦ ♦ tr♥s♠ttr ♦r♥ rr ♣♦st♦♥ s♥tr s sq♥ ♦ ♥trt♦♥ ♥ rt ss r②♥ ♦♥② ♦♥ t♥r♦♥♠♥t t♦♣♦♦② ♥ ♥♦t ♦♥ s♣ ♣♦st♦♥s ♥ tt ♥r♦♥♠♥t ♥ s♥ s♥tr ♥ s t♦ ♣r♦ ♠t♣ r②s ♦r r♥t tr♥s♠ttr♥ rr ♣♦st♦♥s ♦r ts rs♦♥ s♥trs r ♥tr ♥t ♦r♥r♠♥t ♣r♦ss♥

♠t♣ r♣ sr♣t♦♥ s ♥ ♥tr♦ s ♦r ♦ t t♦ sr♦t t s②♥tt ♥r♦♥♠♥t ♥ t♦ ♠♦ t rt♦♥ t♥ ♣♦ss tr♦♠♥t ♥trt♦♥s r♣ sr♣t♦♥ ♣r♦s ♥tr r♠♦r♦t ♦r t s♥tr ♥ ♦r t ♠♦t② ♥ t s♣ sq♥ ♦ ♥trt♦♥s sr♥ t s♥tr s s♣ ♣t ♥t♦ r♣ ♦ ♥trt♦♥ s t r♥t r♦♦♠ st ② ♥ ♥t ♥ ♠♦t② r sq♥ ♦ s♣r♣s ♥♦s

t♦r③ ♣♣r♦ ♦ t t♦♦ s ♥ ♣r♦♣♦s t♦ rs♦ t ♥t♦♥st ♥s ♦ s♣ ♥ r ♥t rqr ② t s♠t♦♥ ♦ ♠♦t②♥ ❯❲ rs♣t② ♣r♦♣♦s s♦t♦♥ s t♥ ♥t ♦ t t♠♥s♦♥ rr② strtrs s r ①t♥s♦♥s ♦ ♠trs ♥ ♠♥s♦♥s ♦♣t♠③ ♦r ♦♠♣tr ♣r♦ss♥ ♥ t ♥trt♦♥s ♦♥ts ♥ t ♥tr ♥♥ ♥ ♦♠♣t ♦r ♥ ♥tr rq♥② r♥ ♥t s♠ t♠ sr♣t♦♥ ♦ t♦s ♦♣rt♦♥ ♥ ♠♥s♦♥s s rqrt♦ ♥tr♦ ♥ ♣t ♠t♠t ♦r♠s♠

♣tr ♦♥s♦♥

st s♣ ♠t ♣♣r♦ s ♥ ♣rs♥t t♥ ♥t ♦ tt♦r③ s sr♣t♦♥ s ♥ t t♦♦ tr♠♥st sr♣t♦♥ ♦t ♥r♦♥♠♥t s s t♦ ♣r♦ ♦ ♥ ♦srs ♣♥♥ ♦♥ t ♣♦r③t♦♥ ♥ ♥♥ ♥ ♥ t ♠t♦ ♥ ♦♥sr s ♦♠♣tt♦♥ ♠t t♦ t rt ♣t r② ♥ t♦♥ ♦ r② ♠t♦♠♣tt♦♥ s♣ t ♠t♦ s s♦♥ ♦♦ ♣r♦r♠♥s ♥ ♦♠♣rs♦♥ tr ♠sr♠♥ts

P ♥ tr♦♥♦s ♦③t♦♥ ♦rt♠ s ♦♥ ♥tr ♥②ss ♣♣r♦ s ♥ ♣r♦♣♦s ♦rt♠ ss t P s t♦ ♠t r♦♥s♦ ♥ s♣ ♥trst♦♥ ♦ t♦s r♦♥s r t s♦t ♣♦st♦♥ s ss♠t t♦ t ♥♦s♥ ♦① ♦r ♦♠♣tt♦♥ ♥② rrsst♦♥ ♣r♦ss ♦♥ tt ♦① s ♣r♦ t♦ ♠t rstrt r r t ♣♦st♦♥ ♥ st♠t t rstrt r ♥ rt② rt t♦ qt②♥t♦r ♦ t ♣♦st♦♥ st♠t♦♥ ♥♦r t♦ t rr♦r r♥ ♦ t st♠t♦♥♥ s♠t♦♥ ♥ ♥ ♠♦st r ss P s ♦t♣r♦r♠ st♠t♦r ♥tr♠s ♦ ♣♦st♦♥ r②

♥ ♦r♥ ♠t♦ t♦ rs♦ t ♦③t♦♥ ♥ stt♦♥s r ♦♥② rt r♥ ♦srs ♥ sr ♣♦r ♦srs r ♣r♦♣♦ss♦t♦♥ s t ♣♦r ♦srs ♥♦t t♦ ♣rt ♦ t ♦ ♦③t♦♥ ♣r♦r t ♦♥② t♦ t♥ t ♣♦t♥t s♦t♦♥s rt ② t r♥♦srs ♥ ♥t ♦ t ♠t♦ s t♦ ♦♥sr t r② ♦ tr♥ ♦sr ♣♦st♦♥ st♠t♦♥ ♥ ♦ ts rt♦♥ ② t ss rt ♣♦r ♦srs ♠t♦ ①♣♦t ♥t♦s② t P ♦rt♠♥ ts ♥tr♥s t② t♦ tr♠♥ ♠t ♠♦ s♦t♦♥s ♣♣ ♦♥ ♠srt t ♠t♦ s s♦♥ ts ♥trst ♦♥② ♥ t ♠t s r r♥ ♦srs ♥ ♥q ♣♦r ♦sr s r♥ t ♠♥ ♦ ♣♦st♦♥♥ rr♦r ♦♠♣r t♦ ♣♣r♦

②♥♠ ♣t♦r♠ s ♥ ♥tr♦ ♥♥ ♠♣♦rt♥t s♣ts ♦r ts♠t♦♥ ♦ ②♥♠ tr♦♥♦s ♥ ♦♦♣rt s♥r♦s rst t rst ♠♦t② ♦ t ♥t s s♠t s♥ srt ♥t s♠t♦r ♥t♠♦t② s sr t r s ♥ t s♠ s ♦r tr♠♥♥ t ♣tt♦ r st♥t♦♥ ♥ ♥t♥ ♥ ♦♥ ♦st ♦ t ♥r♦♥♠♥trs♣t② ♦♥ r♦ ♣r♠trs ♦ ♥ts ♥ t ♦r♥③t♦♥ ♦ tr♦ ♥t♦r s ♠♦♥t♦r ② ♥t♦r r♣ s sr♣t♦♥ ♦ t s♠t♦♥ ♦ ♦t tr♦♥♦s ♥ strt ♥t♦rs ② ♠♥♥ ♠t♣ ♣r ♥t ♥ ② ♥ t♦ ♥t rstrt ♥ s♥tr s♦♥ ♦ t♦ ♥t♦r tr♦ t s♦ ♣rs♦♥ ♥t♦rr ♠♥♠ ♦♠♠♥t♦♥ ♣r♦t♦♦ ♥♥ t s♠t♦♥ ♦ ♦♦♣rt ♦③t♦♥ s♠s s♥ ♠♣♠♥t s tr trs ♥ ♠♦♥strt ♥t♦ s②♥tt♥r♦♥♠♥t ♦r ♥ tr♦♥♦s ♥ ♦♦♣rt ♦③t♦♥ s♥r♦

sr♣t♦♥ ♦ t ②♥♠ ♣t♦r♠ s s♦♥ ♦ ♥♦s ♠♦t② ♥ ♠♣♠♥t ♥ ♣r♦♣t♦♥ t♦♦s ♥ ♦ t ♥ s ♦r ♦③t♦♥ s t ♥t♦r ♦r♥③t♦♥ s s♦♥ ♦ ♣r♦♣t♦♥ t♦♦s ♥ srt ♥ts♠t♦♥ ♦ ♠♦t② ♥ ♥ t♦ ♣r♦ rt Ps ♥ t♦♥ t

Prs♣ts ♥ tr ❲♦r

s♠♣ ♦♠♠♥t♦♥ ♣r♦t♦♦ ♥ ♠♣♠♥t ♦ t♦ tt ♣r♦r♠♥s ♦ strt ♦③t♦♥ ♦rt♠s ♥② t ♦r ♣rs♥t ♥ ts tss s ♥ ♠♣♠♥t ♥ s t ♦r ♦ t ♦♣♥ s♦r♣r♦t P②②rs ❬❪ ♣r♦♣♦s t♦♦ t ♠♦t② ♥t♦r ♥ ♦♠♠♥t♦♥ ♠♦ ♥ t ♦③t♦♥ ♦rt♠ sr ♥ r♥t ♣tr♥ s ♦r r ♦♥ t st

Prs♣ts ♥ tr ❲♦r

♣r♦♣♦s t♦r③ ♦♥t ♥t ②t ♦ t ♥tr ♣♦t♥tts ♦ ts♥tr sr♣t♦♥ ♦ ss t♦ ♥r♠♥t t ♦ ♥ssr②t♦ tr♠♥ ♥ ♦rt♠ t♦ ♣t s♥trs ♥ ♥ t tr♥s♠ttr ♥♦r t rr ♠♦s sr s♥trs r♠♥ ♥ s♦♠ s♣♣rr s t♥ ♥ ♥trst ♥ ♥♥ t st ♣r♦r t♦ r♣ t ♥ss♥trs ② ♥ ♦♥s t rqrs t♦ ♥tr♦ ♥t♦♥ ♦ st♥t♥ s♥trs

rr♥t rs♦♥ ♦ t t♦r③ t♦♦ ♦ ♥♦t t ♥t♦ ♦♥t t rt♦♥ ♥trt♦♥s ♦r t rqr sts ♦ t r♦♠ t s♥trtr♠♥t♦♥ t♦ t ♥trt♦♥ t♦♥s r r② r② t♦ ♥trt trt♦♥s s tr ♥trt r② s♦rt② s t ♥trt♦♥t♦♥ r♥trs ♥t♦ s②♥tt ♥r♦♥♠♥t s s♦ ♦♥sr

t♦r③t♦♥ ♦rts ♣r♦r♠ ♦♥ t ♦ r② sts②♥ s♣♠♣r♦♠♥t t t t♦r③ ♣♣r♦ ♥ ♥tr② ♣r③ ♣r♦ss♥♠r② t st♦♥ ♣r♦ss ♦ t P t ♦rt♠ ♦ s♦ ♥t ♦ ♣r③ rs♦♥

P ♦rt♠ ♦ ♥t♦s② ♠♦ ♥t♦ strt ♥ ♦♦♣rt ♦rt♠ r♦♠ t strt ♣♣r♦ t ♦ ♥s tt ♥♦ ♦♠♣ts ts ♦♥ ♣♦st♦♥ t s♦ t♦s ♦ ts ♥♦rs ♥ ♦rr t♦ tt♣♦t♥t rr♦r ♦r s ♣♥ qst♦♥ ♦r ♦♦♣rt ♣♣r♦ ♦ ♦♥ t♥tr ♦ ①♥ ♥♦r♠t♦♥ t♥ ♥♦s ♦ t ♥♦ sr tr♦①s st♠t♦♥ ♦r ♦♥② tr P ♥ ♦ ts ♦ ♠♣t ♦♥ t ♣♦st♦♥st♠t♦♥

♦③t♦♥ ♦rt♠ s ♦♥ t ②♣♦tss tst♥ ♦ ♥r③♦t ♦r stt♦♥ r ♠♦r tt ♠t♠♦ s♦t♦♥s ♣♣r s t②♣♦tss tst s♦ tr♠♥ ♦r r♥t rr♦r ♠♦ ♦ r ♣♦r♦r r♥s

♠♦t② st ♦ t ②♥♠ ♣t♦r♠ ♦ st r♦♠ ♥t r♣rs♥tt♦♥ t♦ ♦② r♣rs♥tt♦♥ ♥ ♦rr t♦ rss ♣r♦♠ts ♥t♦♥ s r♣rs♥tt♦♥ ♦ ♦ t ♥♥ ♦ t ♣♦st♦♥ ♦ t♥t♥♥ ♦♥ t ♦② ♥ t ♥♥ ♦ t ♦② ♦♥ t ♣r♦♣t♦♥ ♥♥❲t s ♥ ♦t♦♥ t ②♥♠ ♣t♦r♠ ♦ ♦♠ ♥t ♦r ②♥♠ tr♦t tt♦♥ s♠t♦r

t s ♠♦ ♦ ♥t♦s② s s st♥ ♦♥ ♣♣t♦♥

♣tr ♦♥s♦♥

♦r ♥ ♠tr rtr③t♦♥ ♥ ② ♦♣♥ t♦ r ♠sr♠♥ts ♥r♥♥ ♦rt♠s t s♦ ♣♦ss t♦ tr♠♥ t r♥t ♣r♠trs ♦ ♥ s

♥② t s ♦♠♠♥t♦♥ ♣r♦r ♠♣♠♥t ♦ ♥t r♦♠ t♥ ♥t♦ ♦♥srt♦♥ r ②r ①♥ ♣r♦t♦♦s ♥♦t② ② ♥tr♥t t t ♣r①st♥ ❲ t♦♦ ❬❪

♣tr

és♠é ♥ r♥çs

♥tr♦t♦♥

♣s s r♥èrs ♥♥és ♠♦♥ s ♦♠♠♥t♦♥s s♥s r♥♦♥tr ♥♦♠r① és Pr♠ ① ♥ s ♣s ♠♣♦rt♥ts st tr♦r s s♦t♦♥s♣r♠tt♥t ss♠r ♥ r♦ss♥ ①♣♦♥♥t ♠♥ ♥ tr♠ ♦♥♥ést♦t ♥ rés♥t ♦♥s♦♠♠t♦♥ é♥rétq ♥ rés♦r s ① ♠♥s♥t♦♥sts s ér♥ts ♣sts ♥sés t♥♥t à r♥r s rés① s♥s ♣s♣tts à rs ♥r♦♥♥♠♥ts t ① rss♦rs s♣trs s♣♦♥s ♥ ♣rtr s t♥qs r♦ ♥t♥ts ♦ rés① t♦ ♦r♥sés P♦r ② ♣r♥r♥ s ♦♥t♦♥s ♥éssrs st ♦♥♥ss♥ ♣♦st♦♥ s tr♠♥①

♥ t ♥♦r♠t♦♥ ♣♦st♦♥ q ♦♥t♠♣s été ♥ r ♦té à st♥t♦♥ s tstrs ♣♦rrt ♥r ♥ t♦t ♣♦r s ♦♣értrs téé♣♦♥qs♥s ♦♥♥ss♥ ♣♦st♦♥ s tr♠♥① ♣r♠ttrt à ♦♣értr ♦♣t♠srs♦♥ rés ♥ tr♠ ét t ♦♥s♦♠♠t♦♥ é♥rétq tt tst♦♥ ♣♦st♦♥ ♥s ♦♠♥ s téé♦♠♠♥t♦♥s t rs ♦t ♥♦♠rssrrs ♥♦t♠♠♥t ♥ r♦♣é♥ s ♣r♦ts P ❲ t P ❲ rr s ér♥ts rrs ♣♣rît ♥trtrs q t ♦r♥st♦♥ ♥trsé s rés① ♦♠♠♥t♦♥ ♣♦rrtêtr r♠s ♥ qst♦♥ ♥ t ♠t♣t♦♥ ♥♦♠r tr♠♥① ♠♦s ♥♦♠r st♥rs s♣♦♥s sr ♥ ♠ê♠ tr♠♥ t ♦♥♥ss♥ s ♥♦r♠t♦♥s ♦st♦♥s ♦r♥t ♦ ① rés① strés

Prés♠♥t q s trs rés① s♥s s sr♦♥t sés sr ♥ rttr stré t étér♦è♥ rr♦♣♥t ér♥ts st♥rs t q ♥♦r♠t♦♥ ♣♦st♦♥sr ♣r♦♠♥t ♥ s és ♣♦r tt é♦t♦♥ st ♥éssr ès ♠♥t♥♥t♥sr s ♠ét♦s t s ♦rt♠s ♦♣t♠sés ♣♦r s é♦t♦♥s ét ♣♦♦r r s ♠ét♦s sr s s②stè♠s rés st ♥éssr ♦♥strr ss♠trs ♣r♠tt♥t rssr s ♥♦s ♣r♦é♠tqs P♦r r st♥éssr ♦♠♣r ♣srs tâs s♠t♥é♠♥t ♦t ♦r ♥ êtr ♦♠♣t s trs st♥rs ♦♥t s réq♥s t rr ♥ s♦♥t ♥♣r♣éts ♠♥tt♦♥s ♥ t s♠tr ♦t ♣r♠ttr résr s s♠t♦♥s

♣tr és♠é ♥ r♥çs

♣♦r s s②stè♠s tr r ♥ ♥ ♦♠♣♥r rs é♦t♦♥s ♣réss♥st ♠t♣t♦♥ s tr♠♥① s♥s st ♣r♥♣♠♥t é à r s♣t♠♦ ♥s ♠♦té st ♥ r ♠♣♦rt♥t à ♣r♥r ♥ ♦♠♣t ♥s ♥s♠tr t s♣t s♠t♦♥ ♣♦s t♦t♦s ① és ♠rs rér ♥♠♦è ♠♦té rést ♥ ♣r♦r s ♦srs s♠és ♥ r♣♣♦rt rété ♥térr ♠♦té ♦♠♠ ♥ ♣r♠ètr à ♣rt ♥tèr s♠t♦♥ ♥♠♥t ♦♥sttt♦♥ s rés① strés st ♦♥t♦♥♥é ♣r ♣té q♦♥ts ♥÷s rés à ♣♦♦r ♦♠♠♥qr ♥tr ① té ♥st ♣s t♥t ♦♠♠♥t♦♥ ♥ t♥t q t ♠s s strtés à ♠ttr ♥ ♣ ♣♦r q tt♦♠♠♥t♦♥ s♦t ♦♣t♠sé r tt qst♦♥ ♥ét♥t ♣s ♥♦r tr♥é ♣r ♦♠♠♥té s♥tq st ♥éssr rér ♥ s♠tr ♣r♠tt♥t ①♣♦rrt ér s ér♥ts ♦♣t♦♥s ♣♦sss

♥ s ♦ts ♣r♥♣① tt tès st ♦♥strr ♥ ♣t♦r♠ s♠t♦♥ ♣r♠tt♥t ré♣♦♥r à t♦ts s ♥trr♦t♦♥s ♦♥strt♦♥ ♥t ♣t♦r♠ s♠t♦♥ ♥ésst s♥térssr à ♠♦ést♦♥ ♣r♦♣t♦♥ à ♥②s s é♣♠♥t s ♠♥s ♥s ♥ ♥r♦♥♥♠♥t ♥térrt ① ♣r♦é♠tqs ♦st♦♥ tt tès ♣r♦♣♦s ♥s rr ♣r♦♣t♦♥ t ♠♦té ♠♦té ① ♦rt♠s t s ♦rt♠s à ♣r♦♣t♦♥ ♣r♥tr♠ér ♥ ♥q ♦t s♠t♦♥

P♥ ♦♠♥t

♦♠♥t st sé ♥ ♣trs s ♣r♠rs ♣trs t rss♥t ♥ ♣♦♥t s♣éq ♣t♦r♠ s♠t♦♥ ♣r♦♣t♦♥ ♦st♦♥s♠t♦♥ ②♥♠q ♥ s ♣trs ♦♥t♥t ♥ sr♣t♦♥ ♣♦♥t♦ré ♥s q♥ ♣rt t♦♥ r♥èr st♦♥ st éé à t♦♥s ér♥ts ♠ét♦s sr s ♦♥♥és rés sss ♥ ♠♣♥ ♠sr

♣r♠r ♣tr ♣rés♥t ♥ ♦t tré r②♦♥ t♦rsé sé sr ♥sr♣t♦♥ ♥r♦♥♥♠♥t à r♣ ♦t ♦r s ér♥ts r♣s tsés ♣♦r ♠♦ést♦♥ ♥r♦♥♥♠♥t s♦♥t ♣rés♥tés s r♣s s♦♥t à ♦r♥ ♥tr♦t♦♥ ♦♥♣t s♥tr ♣r♠tt♥t sr♣t♦♥ s ♦♠♣♦s♥tssttqs ♥ s s♥trs s r②♦♥s ♣s s♦♥t ♦t♥s ❯♥ ♦ss r②♦♥s ♦t♥s ♦t♥t♦♥ ♥ ♣r♦♣t♦♥ t tr♥s♠ss♦♥ à ♣rtr sr②♦♥s st ért à ♥ ♦r♠s♠ t♦rsé s♣♣②♥t sr s strtrs t ♠♥s♦♥ rr②s s tt t♦rst♦♥ ♣r♠t é♠♥t ♥tr♦t♦♥♥ ♠♦è ♠t ♥ r♥èr ♣rt ♣tr

s♦♥ ♣tr ♣r♦♣♦s ♦rt♠s ♦st♦♥ ♥ ♠ ♥térr♦t ♦r ♥ ♠ét♦ ♦st♦♥ sé sr s ♠ét♦s ♥②s♥trs st ért tt ♠ét♦ ♣♣é P st ♥st é ♥ s♠t♦♥t ♦♠♣ré à trs ♣♣r♦s t②♣ ♥s ♥ s♦♥ ♣rt ♦rt♠ P st ♠s à ♣r♦t ♣♦r ①♣♦tr ♥ ♠ét♦ ♦st♦♥ sé sr ♥ tst②♣♦tès tt ♠ét♦ st é♠♥t é ♥ s♠t♦♥

tr♦sè♠ st♦♥ rss ér♥ts s♣ts ♣t♦r♠ ②♥♠q ♠♦té ♦r♥st♦♥ rés t s ♦♠♠♥t♦♥s ♥tr ♥÷s ♠♦té s

♦ést♦♥ étr♠♥st ♣r♦♣t♦♥ ❯♥ ♣♣r♦ t♦rsé sé srs r♣s

♥ts st s♠é t♦t ♦r à r♥ é ♥ ts♥t s r♣s ♣♦r é♥rs ♦ts t s trt♦rs s ♥ts ♣s à ♣tt é ♥ ts♥t ♥ ♠♦è str♥ ♦rs ♣♦r érr é♣♠♥t t ét♠♥t ♦sts ♦r♥st♦♥ rés st ♠♦ésé à ♥ r♣ ♣♦r srr s ♣♦t♥ts é♥s ♥tr♥ts ♥♠♥t ♥ ♣r♦t♦♦ ♦♠♠♥t♦♥ ♥tr ♥ts ♠♥♠ st ♥tr♦ts tr♦s s♣ts s♦♥t ♥st tstés ♥s ♥ ♣rt s♠t♦♥

r♥èr st♦♥ s♥térss à t♦♥ s ♦rt♠s ♥ ts♥t s♦♥♥és sss ♥ ♠♣♥ ♠sr ♣r♦t P ❲ s ♦srsr♦ sss ♠t t s ♦t♥s ♣r s♦♥t ♦♥r♦♥tés ① ♠srs ♠ê♠ ♣rés♦♥ st♠t♦♥ ♣♦st♦♥ s ① ♦rt♠s ♦st♦♥ stéé ♥ ts♥t s ♦♥♥és ♠srés

♦ést♦♥ étr♠♥st ♣r♦♣t♦♥ ❯♥ ♣♣r♦ t♦rsé sé sr s r♣s

♥ ♣♦♦r rssr s ♣r♦é♠tqs s rés① é♠r♥ts ♦♥t ♥sért à q ♠♦♥té ♥ réq♥ t ♦♥t ♠t♣t♦♥ s tr♠♥① ♥é q rs ♣r♦♣♥s♦♥s à êtr ♠♦s ♥t ♥éssr rér s ♦ts s♠t♦♥ r♣s t ♦♣t♠sés

tt ♣rt ♣rés♥t ♥ ♦t tré r②♦♥ ②r♥ sé sr sr♣s t sr ♥ ♣♣r♦ t♦rsé ré♣♦♥s ♠♣s♦♥♥ ♥♥♥ ♠♣s s♣♦♥s tst♦♥ s r♣s rt ♣r♠ttr ré♣♦♥r ① ♣r♦é♠tqs ♠♦té ♥♦t♠♠♥t ♥tr♦t♦♥ ♦♥♣t s♥tr q ♣r♠t r é♥érr r♣♠♥t s r②♦♥s ès ♦rs q é♠ttrt♦ ré♣tr ♥ s s♦♥t q ♣ é♣és ♣♣r♦ t♦rsé ♣r♠t q♥t à ♦♣t♠sr ét♦♥ ♣♦r ♥ très r ♥ réq♥ ♥♥ t♦rst♦♥ ♣r♠t é♠♥t é♦♣♣♠♥t ♥ ♠♦è ♠t s♣éq♣r♠tt♥t ér r♣♠♥t à s Ps

tst♦♥ s r♣s ♥s ♦t

tt ♣rt s♥térss à sr♣t♦♥ ♥r♦♥♥♠♥t ♦t à r♣s

sr♣t♦♥ ♥r♦♥♥♠♥t ♣r s r♣s

♥ érr ♥r♦♥♥♠♥t s♠t♦♥ r♣s s♦♥t ♥éssrs r♣ strtr Gs(Vs,Es) t r♣ t♦♣♦♦ Gt(Vt,Et) r♣ strtrGs ♦♥t ♥ ①♠♣ r♣rés♥tt♦♥ st ♦♥♥é ♥ r ♣r♠t ♠♦ésrs ér♥ts ♣rts ♥ ♣♥ ât♠♥ts ♥s ss ♥÷s ér♥t à ♦s ss♠♥ts t rêts s éé♠♥ts ♣♥ r♣ t♦♣♦♦ Gt r♣rés♥té ♥ r st ♦♥strt à s s ②s Gs ❯♥ ② st é♥ ♣r ♥ ♠♥ r♠é♣rt♥t t ♥ss♥t ♥ ♠ê♠ ♥÷ t q ♥♠♣r♥t q♥ s ♦s s ♥÷s


r♣ strtr

0

1 23456

7

8

910111213

14 15

16

r♣ t♦♣♦♦

P♦r ♥ ♠ê♠ ♥r♦♥♥♠♥t ♥ ①♠♣ r♣ strtr Gs t r♣ t♦♣♦♦ Gt

t s ♥s ♥s ♠♥ tt r♣rés♥tt♦♥ ♥ ② ♣r♠t ♥ éért♦♥ trt♠♥t t rét♦♥ trs r♣s ts ♣♦r ♣r♦♣t♦♥

s r♣s ♣♦r étr♠♥r s s♥trs r②♦♥s

♦t ♣r♦♣♦sé sé sr ♦♣tq é♦♠étrq t té♦r ♥♦r♠ rt♦♥ ❯ r♣♦s é♠♥t sr ♥ sr♣t♦♥ s r♣s ♣♦r érr ♣r♦♣t♦♥ s ♦♥s étr♦♠♥étqs P♦r ① ♥♦① r♣s s♦♥t♥tr♦ts

r♣ sté Gv

r♣ sté Gv ♣r♠t érr sté ♦♣tq ♥tr s ♣♦♥ts ts rêts ♣♥ ♥ ât♠♥t s ♥÷s Gv s♦♥t ♥tqs à ① Gs s♥s Gv r♥t s ♥÷s ♣♦♥ts ♦ s♠♥ts ♥ sté ♦♣tq r r♣rés♥t ♥ r♣ sté

1

2

3

4

5

6

7

8

9

10 11 12 13 14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42 43

44

45

46

47

48

49

50

51

52

53 54

55

5657

5859

6061

6263

6465

66 67

6869

7071

7273

7475

76 77

7879

8081

82 83

8485

8687

88 89

-71 -70

-69 -68

-67 -66

-65 -64

-63 -62

-61 -60

-59 -58

-57 -56

-54

-53 -52

-51 -50

-49 -48

-40-23

-22

-2

-9-8

-7-6

-5-4

-3 -1

r♣ ❱sté Gv

r♣ sté st ♥ ét♣ ♥tr♠ér ♣♦r étr♠♥r ♥ r♣♥trt♦♥s ♥ t r♣ ♥st ♣s ss♥t ♥ ♣♦♥t étr♦♠♥étq ♥st ♣r ①♠♣ ♣s ♣♦ss s r♣ r st♥t♦♥♥tr ♥ ré①♦♥ t ♥ tr♥s♠ss♦♥ ♥s r♣ Gi st ♥tr♦t ♣♦r r tt♠ïté


r♣ ♥trt♦♥s Gi

r♣ ♥trt♦♥s Gi st ♥ r♣ ♦r♥té q rss♠ t♦ts s ♥trt♦♥s♣♦sss ♥tr s ♥÷s Vs q ② Gt ♥ ♣rtq ♥ ♥trt♦♥ Gv♣t é♥érr 4 ♥trt♦♥s ♥s Gi ré①♦♥ ♥ ♦té t tr ♥ s t♦ tr♥s♠ss♦♥ ♥s ♥ s♥s t ♥s tr ♥ s ♥♠♥t ♦♥ ♦♥sèr t②♣s♥trt♦♥s tr♥s♠ss♦♥ ré①♦♥ t rt♦♥ r str s ér♥ts rt♦♥s ♥tr s ♥trt♦♥s r♣ Gi

-5

1

2

3

4

5

6

7

0 1

-68

9

10

11

12

-7

-8

2

-1 -2

-3-4

-10

-9

(7,1)

(8,1)

(9,2)(9,1)(9,1,2)

(9,2,1)(-8)

(-7) (10,2)

(11,2)

(12,2)

(5,1)

(6,1)

(4,1)

(4,0)

(4,0,1)

(4,1,0)

(2,0)

(1,0)

(3,0)

sss s♦♥t r♣rés♥tés s ♥÷s Gs t s ♥♠ér♦s ②s ♥s s rrés

r♦s Gt ♥ ss♦s st r♣rés♥té r♣ ♥trt♦♥ Gi ss♦é s ♥÷s♦rrs♣♦♥♥ts à ♥ ré①♦♥ s♦♥t ♥ ♥s à ♥ tr♥s♠ss♦♥ s♦♥t ♥ ♦r♥s t à

♥ rt♦♥ s♦♥t ♥ rt

s r♣s ① s♥trs r②♦♥s

♥tr♦t♦♥ ♦♥♣t s♥tr st ♠♦té ♣r ♣srs ♣♦♥ts ♦t♦r ♦srt♦♥ s ré♣♦♥ss ♠♣s♦♥♥s ♥ ♠♦♥tr q ①sts ♠♥s ♣rsst♥ts ❬❪ ♥ r ♦rsq♥ tr♠♥ t ♥ ♣tt é♣♠♥t♥s s♣ s ♦♥trtrs ré♣♦♥s ♠♣s♦♥♥ s♦♥t ♦♠♥t s ♠ê♠sq♥t é♣♠♥t ♣s s ♦♥trtrs s♦♥t ♥♥és ♣r t♦♣♦♦ ♥r♦♥♥♠♥t t ♣r s ♦♠♣♦st♦♥ ♠tér① tsés ❯♥ s♥tr r♣rés♥t♥ ♦♥trtr ♥ Pr ①t♥s♦♥ ♣rt stt♦♥♥r st ♦t♥ ♥ ré♥t s ér♥ts s♥trs tr ♣rt été ♠♦♥tré q ♠♦ést♦♥ st♦stq ♥ t ♣s ♣rtèr♠♥t ♥♥ rérért♦♥ sr ♣♦t êtr ♠♦ésé à r♣s ❬❪ Pr ①t♥s♦♥ st♦♥tré q st ♣♦ss r♣rés♥tr ♦ s ♦♥trtrs ♠♣♦rt♥ts ♥ ♥ ts♥t s r♣s


♥t♦♥ ②♦♥ ❯♥ r②♦♥ st ♥ ♣♦② ♥ ♦♥t q s♠♥t sté♠té ♣r ♥ ♥trt♦♥ ♦ ♥ ♣♦♥t ♥s ♥ r♣èr (O, x, y)❯♥ r②♦♥ r Lh ♣♦♥ts ♥trt♦♥s st ♥ st ♦r♦♥♥é Lh + 2 ♣♦♥ts ♥ ♦♥sér♥t p−1 tpLh

q s ♣♦♥ts ①tré♠① st q r♣rés♥t♥t é♠ttr t ré♣tr

r = [p−1, p0, . . . , pLh−1, pLh]

♥t♦♥ ♥tr ❯♥ s♥tr r②♦♥ ♦ s♥tr s = S(r) Lh ♥trt♦♥s st ss♦é à séq♥s ♦r♦♥♥és ♦♥rs Lh ♣r♠èrséq♥ st ♥ st ♥q♥t t②♣ ♥trt♦♥ s♦♥ séq♥ st♥ st ♥q♥t ♥tr s ss♦é à ♥trt♦♥ ❯♥ s♥tr ♣t êtr♦t♥ à ♣rtr r♣ Gi ♦ éré à ♣rtr ♥ r②♦♥

s =

[

n0 . . . nl . . . nLh−1

t0 . . . tl . . . tLh−1

]

s s♥trs s♦♥t ♦t♥s à ♣rtr s ♥÷s Gi ♥ ét♣s ♦t ♦r♦♥ rr s ②s é♣rt t rré ♥s sqs s tr♦♥t rs♣t♠♥té♠ttr t ré♣tr s ②s t♦s s ♥÷s ② é♣rt t ② rré s♦♥t ①trts ♥♥ ♥ rr s ♠♥s r♥t s ♥÷s ② é♣rt à ① ② rré st té s ♠♥s s♦♥t ♥ sss♦♥ ♥÷s Gi ♦ù ♣♦r ♥ ♥tr ① st ♣♦ss ss♦r ♥ t②♣ ♥trt♦♥t ♥tr s r♥♦♥tré ♥ ♦t♥t ♥s ♥ ♥s♠ s♥trs q ♦♥tsrr à ♦t♥t♦♥ s r②♦♥s

s s♥trs ① r②♦♥s

♥ ♦t♥r s r②♦♥s à ♣rtr s s♥trs ♦♠♠ stré ♥ r ét♣s s♦♥t ♥éssrs ♦t ♦r rrr s s♥trs ♦♥♥♥t s r②♦♥s s ♥s ♣♥ (O, x, y) ♥ t s s♥trs s♦♥t é♥s ♥ ② à ♥ trss ♥ ♦♥t♦♥ ♣♦st♦♥ é♠ttr t ré♣tr ♥s rs ♣ès rs♣ts st ♣♦ss q rt♥s s♥trs ♥ é♥èr♥t ♣s r②♦♥s s P♦rétr♠♥r s r②♦♥s ♥ ♠ét♦ à s ♠s s♦r t rétr♦ ♣r♦♣t♦♥ r②♦♥ st ♣♣qé tt ♠ét♦ ♦♥sst à érr q ①tré♠té ♣♦②♥ ♦♥stt♥t r②♦♥ ♥s q s ss♦é à ♥trt♦♥ s♥trst♥t ♦ss r②♦♥s r♠♣ss♥t tt ♦♥t♦♥ s♦♥t ♦♥sérés ♦♠♠ s t s♦♥t ♦♥srés♣♦r ♥ ét♣ térr r str tt ét♣ ♥st ♦♥♥t rtr♦r s r②♦♥s réés s♦ t ♣♦♥ tt rr st ♥é♣♥♥t rr ♣réé♥t r②♦♥s s t st ♥s ♥ ♣♥ rt (O, s, y)r♣rés♥té ♥ r ♥ ts♥t ♥♦ ♥ ♣r♦éé ♠ s♦r st♣♦ss ♥ té♦r é♥érr ♥ ♥♥té r②♦♥s réés s♦ t ♣♦♥ r♥èrét♣ ♦♥sst à ré♦♥r s ♥s♠s r②♦♥s tr♦és ♥s (O, x, y) t ♥s(O, s, y) P♦r t étr♠♥r ♥s ♥ ♣r♠r t♠♣s ♦rr ♣♣rt♦♥ sér♥ts ♥trt♦♥s ♠rs s♦ ♣♦♥ ♣s étr♠♥r s ♦♦r♦♥♥és ♠♥q♥ts


❱sst♦♥

♣r♦t♦♥ ♥s (O, x, y) ét♦♥ s ♣♦♥ts♥trt♦♥ ♥s ① r②♦♥ s

r♣rés♥tt♦♥s ♠ê♠ r②♦♥ ♥s ①s ér♥ts

♥ ♥tr s ♥s (O, x, y, z) P♦r r st ♥éssr tsr ♥♣r♠étrst♦♥ sr ♥ s ① r♦♣s r②♦♥s ♥ t ♥ ♣r♠étr♥t s♥trt♦♥s ♥ r②♦♥ ♥tr 0 t 1 ♥s ♣♥ (O, x, y) t ♥s ♣♥ (O, s, y) ♣♣rît q s ① ♣r♠étrst♦♥s é♥érés s♦♥t ♦♠♣rs ♥s ♦rr s♥trt♦♥s ♥s (O, x, y, z) ♣① êtr ♦t♥ s♠♣♠♥t ♥ ss♠♥t t ♥ ss♥t♣r ♦rr r♦ss♥t s éé♠♥ts s ① ♣r♠étrst♦♥s ❯♥ ♦s t ♦rr rtr♦é st ♣♣qr s♠♣s rès é♦♠étr té♦rè♠ ès ♥ ♦t♥rs ♦♦r♦♥♥és ♠♥q♥ts ♥♠♥t ♦♥ ♦t♥t ♥ r②♦♥ r é♥t ♣r

rr = [pr,−1, ...,pr,l, ...,pr,L]

s ♣♦♥ts ♥trt♦♥s pr,l = [xr,l, yr,l, zr,l]T

①t♥s♦♥ s r②♦♥s

sr♣t♦♥ ♥ r②♦♥ st ♥ st ♣♦♥ts ♥trt♦♥s s ér♥ts♣♦♥ts ♥trt♦♥s st ♣♦ss ss♦r s ♥♦r♠t♦♥s s♣♣é♠♥trs qs♦♥t ♥éssrs ♣♦r ét♦♥ tr ♥ ♣r♦♣t♦♥ ♥ é♥t t♦t♦r s ♠trs s Mr t ♥trt♦♥s Nr q rr♦♣♥t rs♣t♠♥ts ♥♦r♠t♦♥s sr ♥tr s t t②♣ ♥trt♦♥ ♠ê♠ ♦♠♠ strésr r st ♣♦ss étr♠♥r

♣♥ ♦♥ ♥♥t ss ♦♦r♦♥♥és(

ei,⊥r,l , e

i,‖r,l

)

rr♥t sr ♥trt♦♥

pr,l tr♥srs à sr,l−1


♣♥ ♦♥ s♦rt♥t ss ♦♦r♦♥♥és(

eo,⊥r,l , e

o,‖r,l

)

♣rt♥t pr,l tr♥srs

à sr,l♥s ♣♦r ♥ r②♦♥ ♦♠♣t♥t L + 1 ♥trt♦♥s ♦ù s ♥trt♦♥s −1 t L + 1

r♣rés♥t♥t é♠ttr t ré♣tr st ♣♦ss rér s ♠trs s ♣♥s♥♥

Bir =

[

ei,⊥r,−1, e

i,‖r,−1, . . . , e

i,⊥r,L−1, e

i,‖r,L−1

]

♠♥s♦♥ (3× 2L) t s ♠trs Bor s ♣♥s s♦rt

Bor =

[

eo,⊥r,0 , e

o,‖r,0 , . . . , e

o,⊥r,L , e

o,‖r,L

]

♠♥s♦♥ (3× 2L)

P♥ ♦♥ ♥♥ t s♦rt ♣♦r ♥trt♦♥ l

♥ tr♥s♠ss♦♥

♥ ♣r♦♣t♦♥ êtr té à t ♠♥s♦♥rr②s s s s s♦♥t s strtrs ♦♥♥és ♠t♠♥s♦♥♥s ♣r♠tt♥t ♥ ♣r ♦r♥tr très t ♣♦t♥t♠♥t t♠♥t ♣rés


♥ ♣rtq s s♦♥t éq♥ts ① ♥♠♣② ♥rr②s rr ♠♣② P②t♦♥ ❬❪ P♦r ♣r♦è♠ ♥ tr♥s♠ss♦♥ s ♦♣ért♦♥s ♦♥têtr tés s à ♠♥s♦♥s ss ♣♣é ①s

réq♥ f r②♦♥ r ♥trt♦♥ l ♣♦rst♦♥ p ♣♦rst♦♥ q ts ♦ts s ♥trt♦♥s ♣♦rr♦♥t êtr éés ♠♥èr t♦r à

s ♥ ♣♦r t♦ts s réq♥s s♠t♥é♠♥t ♥ ét♥t ♥ ♦ t♥ q srt r♦♥♦♣ s ♦♥♥t♦♥s ♥♦tt♦♥ t s ♦♣ért♦♥s sr ss s♦♥t érts ♥s ♥♥①

r♥st♦♥ t♦r s ♦♥♥és

P♦r ♣♦♦r tsr s s st ♥éssr q s ♦♥♥és sr s ①s ♥ s♦♥t ♦♠♦è♥s r t♦s s r②♦♥s ♦♥t s ♥♦♠rs ♥trt♦♥s ér♥ts q s ♠♣ê♥t êtr st♦és rt♠♥t ♥s s s P♦r ♣r ♣r♦è♠ s♦t♦♥ rt♥ st ♦r♥sr s ér♥ts r②♦♥s ♣r ♥♦♠r ♥trt♦♥s♥s ♥ r②♦♥ q ♥st tr q♥ st Ln ♥trt♦♥s st st♦é ♥s ♥♦ ♥trt♦♥s Ln st ♦rs ♣♦ss érr ♥ séq♥ (L0, ..,Ln, .,LNi−1

) Ni ♦s r②♦♥s ♦ Ln st ♦rs ♥ ♦ ♦♥t♥♥t Rn r②♦♥s Ln ♥trt♦♥stt ♦r♥st♦♥ s ♦♥♥és st stré ♣r ♥ ①♠♣ r


①♠♣ ♦r♥st♦♥ s ♦♥♥és ♣♦r r②♦♥s r②♦♥ r♦ ♦♠♣t ♥trt♦♥s st♦és ♥s Ln ♦ ♥♠ér♦ 0 r②♦♥ rt ♦♠♣t ♥trt♦♥s tst st♦é ♥s Ln ♦ ♥♠ér♦ 1 s trs ♦♥♥és st♦és ♥s s ér♥ts Ln♦s ♦rrs♣♦♥♥t à s r②♦♥s ♥♦♥ r♣rés♥tés sr r


♦t♦♥ s ♠trs ♥

♥s ♣rt été q ér♥ts t②♣s ♥♦r♠t♦♥s ♣♦♥t êtr①trts à ♣rtr ♥ r②♦♥ ♥ ts♥t ♦r♠s♠ s st t♦t ♦r♣♦ss ①♣r♠r s ♠trs ss ♣♦r ♥ Ln ♦ n ♣♦r t♦s s r②♦♥s t♦tss ♥trt♦♥s ♥s

inBr,l,p,x = rs♣

(r,l,p,x)

(

inBr,lp,x

)

onBr,l,p,x = rs♣

(r,l,p,x)( onBr,lp,x)

♦ù x st ♥ ① s♣t ♥s ♦♥ ♣t é♠♥t é♥r ♥ ♥tr ♥t sss ♥tré t s♦rt t q

nBr,l,p,q =onB

x,pr,l

inB

x,pr,l

♠ê♠ s ♠trs Nr t Mr ♦♥t♥♥t s ♥♦r♠t♦♥s ♥tr s ss t t②♣ ♥trt♦♥s ♣♥t s①♣r♠r à t êtr ♥s érts

nNr,l nMr,ls ① r♥èrs ♣r♠tt♥t ♥s rér ♥ ♥trt♦♥ ♦

♥♦té

n,i,s,aAr,l

♦ù i s t a s♦♥t s ♣r♠ètrs t②♣ ♥trt♦♥ ♥tr s t ♥♥♥ és à ♥ s ♥trt♦♥s

t♦♥ s ♥trt♦♥s

❯♥ ♣r♠èr ét♣ ♦♥sst à rr♦♣r t♦ts s ♥trt♦♥s ♠ê♠ t②♣ t ♠ê♠ ♥tr ss P♦r t♥t s q t②♣ ♥trt♦♥ t ♥tr s s♦♥t réés à ♣rtr ♥trt♦♥s ♦♠♣èt n,i,s,aAr,l tt ét♣stré ♣r r st ♥éssr r trt♠♥t ♣♦r q t②♣ ♥trt♦♥t s st s♣éq t ♥ ♣① êtr té sr ♠tr ♥trt♦♥s ♦♠♣èt

❯♥ ♦s tt ré♣rtt♦♥ té ♥ s s st éé ♥♠♥t ♣r♠tt♥t ♦t♥r s ♦♥ts tr♥s♠ss♦♥s ♦ ré①♦♥s ♣♦r t♦tss réq♥s ♦♥sérés ♦ts s s ♥s éés s♦♥t r♦♠♥és ♣♦r rrér ♥ ♥trt♦♥ ♦♠♣èt t éé nAf,r,l,p,q ♦♠♠ stré sr r

♥ ♣r♦♣t♦♥ t ♥ tr♥s♠ss♦♥

ét♦♥ s ♥trt♦♥s ♥s t st ♣♦ss ①♣r♠r ♥ ♣r♦♣t♦♥ ♣r♦♣♦sé ♥s ❬❪ à ♥s ♥ ♣r♦♣t♦♥ ♥tr ♣♦♥ts



rt♠♥t sr ♥trt♦♥s ♦r♥sé ♥ Ln ♦s st♥st ♦♥té♥é t ré♦r♥sé ♥♥ q ♥trt♦♥ st r♥é ♣r t②♣ t♥tr s


❯♥ ♦s éés ♣♦r q t②♣ t ♥tr s s s♦♥t rr♦♣ést ré♦r♥sés ♥s rs ♦r♠s ♥ts


a t b ♣♦r ♥ Ln ♦ ♦♥♥é ♣t sérr

abn Cf,r,p,q = nB

p,q

⋆,r,l[Ln]

0∏

u=ln−1

nAp,q

f,r,l[u] nBp,q

⋆,r,l[u]

P♦r ♦t♥r ♥ ♣r♦♣t♦♥ ♦♠♣t abCf,r,p,q st ♦♥té♥r sLn ♦s ♦♥ ① s r

abCf,r,p,q =|r|

N−1

n=0

abn Cf,r,p,q

♥s ♠♥s♦♥ r st dim(r) =∑N

n=0 n × Rn s ét♣s s♦♥t strés ♣r r

r♠♠ ①♣q♥t ♦♠♠♥t ♦t♥r ♥ ♣r♦♣t♦♥ abCf,r,p,q

♣♦r t♦ts s réq♥s s s ♥trt♦♥s ♦♠♣èts t éés ssés ♥Ln ♦s

♥ tr♥s♠ss♦♥ ♦♠♣t ♣t êtr ♦t♥ râ à

abHf,p,q =abγf,p,q

abαrf,⋆,⋆exp(−2jπff,⋆,⋆τ r)

abγf,r,p,q =−jc4πff,⋆

(

1− bSp,q

f,r

)(

1− aSp,q

f,r

)

abαf,r =bF

p,1

f,r (abℓC

p,q

f,raF

p,1

f,r)p,1

f,r

♦ù aSf,r,p,q t bSf,r,p,q s♦♥t s ♣r♠ètrs s ♥t♥♥s a t b t aFf,r,p tbFf,r,p s♦♥t s ♣r♠ètrs rts à rs ♣♦rst♦♥s t à rs ♥s


♦è t❲ s♣éq

tst♦♥ ♦t ♣r♠t ♦t♥r s s très ♣réss ♣♦r ♥ ①trrs ♣r♠ètrs t♦♣♦ é♣♥♥ts P ♦t♦s ♥ ♦♥t♦♥ s ♣♣t♦♥s ré♣♦♥s ♠♣s♦♥♥ ♦♠♣èt ♥st ♣s ♦ré♠♥t ♥éssr t ♥ st♠t♦♥♣s s♦♠♠r r s Ps ♣t sr ♥s s ♥ s♦t♦♥ st tsr ♠♦è ♠t s♣éq

♠♦è ♠t s♣éq ♣t êtr ♦♥séré ♦♠♠ ♥ ♥é r②♦♥srét s♠♣ trt rt ♥tr é♠ttr t ré♣tr ♥s ♥ tr♥t ♣r♦t ♠ét♦ ét♦♥ s ♥trt♦♥s st ♣♦ss st♠r s ♣rtssr trt rt és à trrsé s ♦sts ♥r♦♥♥♠♥t Pf,p,q

Pf,p,q = −20 log10 | eAf,p,q |

♦ù eAf,p,q st ♠tr ♥trt♦♥s ♦♠♣èt t éé t♦ts s ♥trt♦♥s trt rt

♠ê♠ st ♣♦ss ér ①ès rtr τc(ff , o) ♥tr♦t ♣r trrsés ér♥ts ♠tér① ♣♦r ♥ r♠♣ réq♥ f t ♣♦r ♥ ♣♦rst♦♥ ♦♥ o

τc(ff , o) =

arg( cAf [ff ],p[0],q[0] ), ♦r ♥ ♦rt♦♦♥ ♣♦r③t♦♥

arg( cAf [ff ],p[1],q[1] ), ♦r ♥ ♣r♣♥r ♣♦r③t♦♥

cAf,p,q =

0∏

l=Li−1

i=1,s=Ml,a=θlAl,1f,p,q exp(jd

l,1⋆,⋆,⋆)

♦ù Ml t θl s♦♥t s ss t ♥s ♥♥ ss♦és à ♥trt♦♥ l♥s rtr ♦ sr trt rt τ(f, o) st ♦t♥ ♣r

τ(f, o) = τLOS + τc(f, o)

♦ù τLOS st t♠♣s ♣r♦♣t♦♥ ♥tr é♠ttr t ré♣tr ♥ s♣ r

♦♥s♦♥

tt ♣rt ért ♥ ♦t sé sr s r♣s ♥ ♣rtr été♥tr♦t ♦♥♣t s♥tr r②♦♥ ♦♥♣t ♣r♠t ♠♦ésr stt♦♥♥rté rt♥s ♦♠♣♦s♥ts ré♣♦♥s ♠♣s♦♥♥ és à ♥r♦♥♥♠♥t s♠t♦♥ ♦r ♣♦rt ♦♥♣t ♥ré♠♥t ♥ ét♥t r♥tér s r②♦♥s ♥tr ① ♣♦st♦♥s ♣r♦s s s♥trs s r②♦♥s ♣s ♦♥t été ♦♥strts ❯♥ ♦r♥st♦♥ ♦r♥ s r②♦♥s ♣r ♦s ♥trt♦♥s ♠ê♠s ♦♥rs ♣r♠s ♥ trt♠♥t t♦rsé ♥ ♣rtr ♥♦t♦♥ t♠♥s♦♥ rr② été ♥tr♦t ♥s q♥ ♥♦tt♦♥ s♣éq ♣r♠tt♥t t♦rst♦♥ ♦♠♣èt s s ❯tsé à ♦s ♣♦r ét♦♥ s ♥trt♦♥st st♠t♦♥ ♥ ♣r♦♣t♦♥ t tr♥s♠ss♦♥ tt értr ♣r♠t ♥ t r♣ ♠ê♠ ♣♦r ♥ très r ♠♠ réq♥s r♥t t♦♦rs


♥t tt sr♣t♦♥ t♦r ♥ ♠♦è ♠t s♣éq été ért♣♦r ♣r♠ttr ♥ ét♦♥ r♣ s P

♦st♦♥ ♥ ♠ ♥térr s ♣♣r♦s r♦stst étér♦è♥s

♣tr ♣réé♥t ♣rés♥té ♥ ♦t ♣r♠tt♥t rssr s ♣r♦è♠s ♦♠♣①s ♥♦t♠♠♥t ♥ ♦♥t①t ♠♦té ❯♥ s tst♦♥s ♥sés t②♣ ♦ts st t♦♥ ♦rt♠s ♦st♦♥ ❯♥ ♣r♠èr ♣rt ♣tr ♣rés♥tr ♥ ♠ét♦ ♦st♦♥ étér♦è♥ sé sr s t♥qs ♥s♠sts ♣r sst♦♥s Ps tr♥t ♣rt s ♣♦sstés ♦rts ♣r ts ♠ét♦s t ♥♦t♠♠♥t ♣♦ssté ♦t♥r s s♦t♦♥s ♥ s r♥♦srs ♥ t♥q ♦st♦♥ sé sr ♥ tst ②♣♦tèss sr ♣rés♥té ♥ s ♦rt♠s sr é ♥ s♠t♦♥

sr♣t♦♥ ♦rt♠ ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ P

tt ♣rt ♣r♦♣♦s sr♣t♦♥ ♦rt♠ ♦st♦♥ sé sr s ♠ét♦s ♥②s ♥trs ♥s q s t♦♥ ♥ s♠t♦♥

sr♣t♦♥ s ♦♥tr♥ts é♦♠étrqs

♦t ♦♠♠ s ♦rt♠s ♦st♦♥ érq ssqs t q ① séssr ♦rt♠ ♦st♦♥ é♦♠étrq P ts s ér♥ts ♦srs r♦ ♠ ♦ rr ♦ ♠ r♥ ♦ rr ♦ t ♥ tr♥t ♣♦r st♠r ♣♦st♦♥ ♥ tr♠♥ ♦t♦s ♦♥trr♠♥t à s ♦rt♠s érqs P rés♦r é♦♠étrq♠♥t ♣r♦è♠ ♦st♦♥ ♥ ♠t♥t s ③♦♥s s♣ à s ♦srs r♦s

♥ ss♠♥t t q q P ♣t êtr ss♦é à ♦s à ♥ ♥÷ ♥r♦♥t ♣♦st♦♥ st ♦♥♥ t à ♥ ♠♦è rrr ♦♥t s ♣r♠ètrs ♣♥t êtrr♦ssèr♠♥t ♦♥♥s st ♣♦ss é♠tr s ré♦♥s s♣ s♣éqs ♣♦r♥ s Ps

♥s r ♥ ♦sr t②♣ st♥ r éé ♥tr ♥ ♥÷ ♥r à ♣♦st♦♥ a = [xa, ya, za] t ♥÷ à ♣♦st♦♥♥r à ♣♦st♦♥ b = [xb, yb, zb] ♦♥ ♣térr

r = ‖a− b‖+ δr,

♦ù δr st ♥ tr♠ rt t♦♥♥♥s r ♥ ♦sr t②♣ ♦ ∆ ♣r♦♥♥t ① ♥÷s ♥rs

① ♣♦st♦♥s a a′

♦♥ ♣t érr

∆ = ‖a− b‖ − ‖a′ − b‖+ δ∆,

♦st♦♥ ♥ ♠ ♥térr s ♣♣r♦s r♦sts t étér♦è♥s

♦ù δ∆ st ♥ tr♠ rt t♦♥♥ ♥♥ ♥s r ♥ ♦sr t②♣ ♦rt♠ ♣ss♥ rç ♣t êtr ①♣r♠é râ ♠♦è ♣t♦sss♥t

P (d) = P0 − 10np log10(d),

♦ù P0 st ♣ss♥ rç à ♠ètr t np st ①♣♦s♥t ss♠♥t♣rès ❬❪ st♥ d ♣t êtr ré à ♣ss♥ P ♥ ♥r à ♣♦st♦♥

a ♣r st♠t♦♥ s♥t

d = exp(M − S2) + δP

M =log(10)(P0 − P )

10np

S = − log(10)σX10np

♦ù σ2X st r♥ rrr sr ♣ss♥ rç ♥s tr♠ rrr st♥ ss♦é δP ♣t êtr st♠é

δP =√

(1− exp (−S2)) (exp (2M − 2S2))

♥ s♣♣♦s♥t q s tr♠s rrrs δr δ∆ t δP s♦♥t ss♥s ♥trés t r♥s rs♣ts σ2r σ

2∆ t σ2P st ♣♦ss ♦♥strr s ♥trs ♦♥tr♥t

[Ir] [I∆] t [IP ] é♥s ♣r

[Ir] = [r − γσr, r + γσr]

[I∆] = [∆− γσ∆,∆+ γσ∆]

[IP ] = [P (d)− γσP , P (d) + γσP ]

♦ù γ st ♥ tr st♠♥t q s♥s ♥♦r♠t♦♥s ♣rés sr ♦st à γ = 3♣♦r ♦t♥r ♥ ♥tr ♦♥♥

♥s ♥ s Ps ♦♥♥ à s ♠tt♦♥s é♦♠étrqs s♣ ♣rtèrs q ♣r♥♥♥t ♦r♠

♥ ♥♥ ♥ ♥ s♣èr ♥ ♥tré sr a ♣♦r ♦♥tr♥t st♥t ♣♦r ♦♥tr♥t ♣ss♥

① ②♣r♦s ♥ ♦ ②♣r♦♦ïs ré♦t♦♥ ♥ ♣♦r ♥♦♥tr♥t ♦

sr♣t♦♥ ♦rt♠ é♦♠étrq

♦rt♠ P ♣r♠t ♦t♥r ♣♦st♦♥ ♥ tr♠♥ ♥ ét♣s rés♠és♥s t


sr♣t♦♥ s ét♣s ♥éssrs à st♠t♦♥ ♣♦st♦♥

♦♥strt♦♥ s ♦♥tr♥ts é♦♠étrqs rét♦♥ ♦îts ♦♥tr♥ts s♦♥ s ♦îts ♦♥tr♥ts ♣♣r♦①♠t♦♥ ♣r ♦rt♠ r ré♦♥ ♣♣r♦①♠é st♠t♦♥ ♣♦st♦♥ ♥s ré♦♥ ♣♣r♦①♠é

Pr♠ètrs s♠t♦♥

Pr♠ètrs ❱rs

HP [−1, 1]× [−1, 1] km2

HD [−100, 100]× [−100, 100] m2

HR [−10, 10]× [−10, 10] m2

σr 2.97 mσ∆ 3.55 mσX 4.34 dBnp 2.64P0 −40 dB

♦♥strt♦♥ ♦♥tr♥t é♦♠étrq été ért ♥ st♦♥ rét♦♥ ♦îts ♦♥tr♥t ♦♥sst à tr♦r ♦ît ♥♦♥t ss♦é à ♦♥tr♥té♦♠étrq ❯♥ ♦s s ♦îts ♦t♥s ♦♥ rr r ♥trst♦♥ ♥trst♦♥ s ♦îts é♠t ♥ ré♦♥ s♣ ♦♥t♥♥t ♣♦st♦♥ ♥÷ à ♣♦st♦♥♥r ♥térr tt ré♦♥ ♦♥ ♣♣q ♥ ♦rt♠ sst♦♥ t②♣ r❬❪ ré♦♥ st é♦♣é ♥ s♦sré♦♥s s rêts ♥ s s♦sré♦♥ss♦♥t tstés ♥ s ♦♥tr♥ts é♦♠étrqs st ♥ tst ♥s♦♥ ♥ s rêts s ♦îts ③♦♥ ♦♥♥ ♦♥tr♥t t♦ts srêts s♦♥t à ①térr t♦ts s ③♦♥s ♦♥♥s ♦ît st rté s ss♦♥t ♦♥t♥s ♣r t♦ts s ③♦♥s ♦♥♥s ♦ît st és♥é ♦♠♠ ét♥t♥♦é ♥s s trs s ♦ît st és♥é ♦♠♠ ét♥t ♠ë ♣r♦sss sst♦♥ st ré♣été tért♠♥t sr s ♦îts ♠ës ♥ rtèr rrêt sr ♥♦♠r ♦îts ♦ t s ♦îts ♥♠♥t st♠t♦♥ ♣♦st♦♥ stté ♥ r♥t r②♥tr s ♦îts

t♦♥ ♦rt♠ é♦♠étrq

♦rt♠ é♦♠étrq st éé à ♠ét♦ ♦♥tr♦ sé♥r♦ ♦♥séré st ♦♥♥é ♥ r t s ♣r♠ètrs s♠t♦♥ s♦♥t ♦♥♥és♥s t s ér♥ts ♣♦st♦♥s ♥÷ à ♣♦st♦♥♥r s♦♥t ét♦r♠♥t♦ss ♥s HR ♥s sr s rs ♦♥ ♣t ♦srr q ♥s s ér♥ts ♦♥rt♦♥s P ♣rét sr ♠ét♦ ♦t♦s ♥ ♣♣♦rtést ♣s ♠té tst♦♥ ♦srs ♣ss♥s t ♦♠♣①té àétr♠♥r ♦ît ♥♦♥t ♥ ♦♥tr♥t


♥÷ à ♣♦st♦♥♥r ♥ ♣♦st♦♥ β ès à s ♦srt♦♥s st♥sr ♣r♦♥♥t s ♥rs ① ♣♦st♦♥s AR à s ♦ ∆ ♣r♦♥♥t s ♥rs① ♣♦st♦♥s AD t s P ♣r♦♥♥t s ♥rs ① ♣♦st♦♥s AP

♦st♦♥ étér♦è♥ sé sr tst ②♣♦tèss

P ♣r♠t rés♦r ♣r♦è♠ ♦st♦♥ é♦♠étrq♠♥t ♥ ♠t♥t s ré♦♥s ♥s ♣♥ ♥ ♦ ♥s s♣ ♥ ♥ s ♥ts tt ♠ét♦ rés♦t♦♥ st s ♣té à é♠tr s♣t♠♥t s ré♦♥s ♣s q♥ ♥♦♠r ♦srs r♦s st ♥ss♥t ♠♣♠♥t ♥÷ à♣♦st♦♥♥r ♣t êtr ♠ ♥ ♦rr♥ ♣srs ré♦♥s s♦♥t♥s ssrs♠s s ♥s q s trs ♣♥t ♣♣rîtr P ♦r ♣♦ssté é♠tr s♣t♠♥t t♦ts s ré♦♥s

tr ♣rt ♦♥ r♠rq é♥ér♠♥t q ♣♦st♦♥ st♠é à ♣rtr ♦srs st♥s st é♥ér♠♥t ♣s ♣rés q ♦t♥ ♣r s ♥♦r♠t♦♥s ♣ss♥s ttr st ♦♥tré q s ① ♥♦r♠t♦♥s ♥s r ♦st♦♥ étér♦è♥ ♥ ♣♥t êtr tsés ♠ê♠ ♠♥èr ♠ét♦sé sr tst ②♣♦tès ♣r♦♣♦s t♥r ♦♠♣t tt ss②♠étr qté♥♦r♠t♦♥

sr♣t♦♥ ♠ét♦

sé♥r♦ ♦st♦♥ étér♦è♥ ♦♥séré st ♦♠♣♦sé ♦srs st♥ t ♥ rt♥ ♥♦♠r ♦srs ♣ss♥s ♦r♥s ♣r s ♥÷s♥ts ♦♠♠ r♣rés♥té ♥ r Ptôt q tsr t♦ts s ♥♦r♠t♦♥ssr ♠ê♠ ♣♥ st ♣r♦♣♦sé ♥ ♠ét♦ ♦st♦♥ ♥ ét♣s ♦t ♦r♦t♥r ré♦♥s st♥ts ♥ ♣♣q♥t P s ♦srs st♥s Ps


0 2 4 6 8 10

ˆ

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

♦

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

st♥

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

st♥ ♦

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0CRLB

Geometric

ML-WLS

②rt♦♥ ♦♠♣èt

s rrr ♣♦st♦♥♥♠♥t ♦♠♣r♥t P à s ♦rt♠s t ❲


♥÷ à ♣♦st♦♥♥r ♥ ♣♦st♦♥ B rç♦t ① ♦srs st♥sr1 t r2 s ♥rs ① ♣♦st♦♥s R1 t R2 rrr ♦♠♠s sr s st♥s st♠♦ésé ♣r ① ♥♥① ♦♥♥s ♥trés sr R1 t R2 ♥trst♦♥ s ♥♥① é♥èr s ré♦♥s rsés C1 t C2 rs r②♥trs C1 t C2 ♣s ♥÷ à ♣♦st♦♥♥r rç♦t s ♥♦r♠t♦♥s ♦ ♣ss♥ s ♥÷s ① ♣♦st♦♥s Hk ♥ ♥♦t dk,i st♥ ♥tr Hk t Ci

♦sr ♥tr s ① ré♦♥s à s ♦srs ♣ss♥s♥ ♠♦és♥t ♥♦r♠t♦♥ ♦ ♣ss♥ ♥÷ ♥t k ♣r

Pk = µ(dk) +Xk,

♦ù Xk st ♥ r ét♦r ss♥♥ ♥tré t r♥ σk st ♣♦ss ér ♥tr s ① ③♦♥s réés ♣r s ♦srs st♥s rè és♦♥ s♥t

K∑

k=1

1

2σ2k

[

µ2k,2 − µ2k,1]

C1

≶C2

K∑

k=1

1

σkPk(µk,1 − µk,2).

éstt s♠t♦♥

♥ ts♥t s ♣r♠ètrs s♠t♦♥ ♦♥♥és ♣r t t ① ♥÷s♥ts ♦♥ ♦t♥t ♣rés♥té ♥ r ♥ ♦t q ♠ét♦ rs ♥ ♦rt♠ ♥s s é♦t♦♥ ♥ ♦♥t♦♥ ♥♦♠r ♥÷s ♥ts ♣rés♥tés ♥ r ♠♦♥tr q ♠ét♦ à ♥ ♥térêt♣r♥♣♠♥t q♥ ♥♦♠r ♦srs ♣ss♥s st ♠té


Pr♠ètrs s♠t♦♥

Pr♠ètrs ❱rs

S [−20, 20]× [−20, 20] m2

L [−80, 80]× [−80, 80] m2

σδ 0.9 mσX 4 dBnp 3P0 −40 dB

rrr ♣♦st♦♥ ♥÷s ♥ts

♥ ♦♥t♦♥ ♥♦♠r ♥÷s ♥ts

♣t♦r♠ ②♥♠q érr ♣r♦♣t♦♥ ♦♠♠♥t♦♥ t ♠♦tés ♥ts ♥s rés

♦♥s♦♥

tt ♣rt ♣rés♥té ① ♦rt♠s ♦st♦♥ ♥ t♥q ♦st♦♥ étér♦è♥ sé sr s ♠ét♦s ♥s♠sts P t ♥ ♠ét♦ sé sr tst ②♣♦tèss P ♣r♠t ♥tr trs é♠tr s♣t♠♥t s ré♦♥s s♣ stss♥ts s ♦srs r♦s ♠ét♦ ♠♦♥tré s♦♥ ♥térêt ♥♦♠♣rs♦♥ à s t♥qs ♥ s♠t♦♥ ❯♥ s ♥ts P sts ♣té à é♠tr ♣srs ré♦♥s s♦♥ts ♥♦t♠♠♥t ♥s s s r♥♦srs tt ♣r♦♣rété été ♠s à ♣r♦t ♣♦r é♦rr ♥ ♦rt♠ sé sr♥ tst ②♣♦tèss ♥ t ♥s ♥ s ♦ù ss ① ♦srs st♥ss♦♥t s♣♦♥s P ♣r♠t ♥tt♦♥ s ① ré♦♥s ♥ts ♦♥t♥♥t♣♦t♥t♠♥t ♥÷ à ♣♦st♦♥♥r ♠ét♦ tst ②♣♦tèss ♦♥sst ♦rsà ♦sr ♥tr s ① ré♦♥s à ♦srs ♣ss♥s ♦rs q♥♣♣r♦ ssq ♣♦rrt érr ♣rés♦♥ ♣♣♦rté ♣r s ♦srs st♥s ♥ s s♦♥♥♥t s ♦srs ♣ss♥s ♣♣r♦ ♣r♦♣♦sé ♦♥sr ♠ét♦ ♠♦♥tré q ♣♦t rsr ♥ ♥s s s ♦♣ ♦srs ♣ss♥ s♦♥t s♣♦♥s

♣t♦r♠ ②♥♠q érr ♣r♦♣t♦♥ ♦♠♠♥t♦♥ t ♠♦té s ♥ts ♥s rés

♥ ♣r♦♣♦sr ♥ ♦t s♠t♦♥ ♣♦r ♦st♦♥ ♥ ♥térr ♦♠♣t ♦♥♥t ♦tr à s♠t♦♥ ♣r♦♣t♦♥ ♠♦té s ♥ts t r♣té à ♦♠♠♥qr ♥tr ① tt st♦♥ érr tr♦s s♣ts ♣t♦r♠ s♠t♦♥ ♠♦té ♦r♥st♦♥ s rés① t ♦♠♠♥t♦♥ ♥tr ♥÷s❯♥ r♥èr ♣rt é♠♦♥trr ♥térêt ♣t♦r♠ à trrs s♠t♦♥s ♥①é sr ①trt♦♥ s Ps t tr str♥t ♥ sé♥r♦ ♦st♦♥♦♦♣ért

♦té s ♥ts

♠♦té s ♥ts ♣t♦r♠ ②♥♠q st ssré ♣r ♥ s♠tréé♥♠♥t srt ttr q ♥t èr ♠ê♠ s ♠♦té ♥é♣♥♠♠♥t s trs ♠♦té s ♥ts st ért ♥ ét♣s ❯♥ ♣r♠èr ét♣r é q s♣♣ sr s r♣s ♥ q ♥t ♦sss ♥ st♥t♦♥ tétr♠♥ ♠♥ ♣♦r tt♥r t ♥ s♦♥ ét♣ ♣tt é ♦ù ♥t sé♣ sqà s st♥t♦♥ ♥ ♥trt♦♥ ♥r♦♥♥♠♥t

♦té r é sé sr s r♣s

♥ q s ♥ts ♣ss♥t ♦sr ♥ st♥t♦♥ t tr♦r ♥ ♠♥ ♣♦r s②r♥r s r♣s ♣ès Gr t ♠♥s Gw s♦♥t ♥tr♦ts ♦♠♠ r♣rés♥té sr r s ♥÷s r♣ Gr s♦♥t s ♣ès t s ① ♣ès s♦♥t ♥ts♥ ♥ st réé ♥tr s ① ♥÷s rs♣ts r♣ ♣ès ♣r♠t ♦♥strr


r♣ ♠♥ r♣rés♥té ♥ r r♣ ts s ♥÷s r♣ ♣ès ①qs s♦♥t ♦tés s ♥÷s ♦rrs♣♦♥♥t ♥tr s ♦♦rs t s♣♦rts s ♥÷s s♣♣é♠♥trs ♣r♠tt♥t é♥r s ♦ts ♥tr♠érs① ♥ts t ♥s sr ♥ trt♦r ♣s rést s ♥s r♣ s♦♥t♦♥strts s♥t ① rès

s ♥÷s ss♦és à ♥ ♣è s♦♥t rés ① ♥÷s ss♦és ① ♣♦rts tt♠ê♠ ♣è

① ♥÷s ss♦és à s ♣♦rts s♦♥t rés ♥tr ① s s ① ♣ès ①qss ♣♣rt♥♥♥t s♦♥t ♥ts

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strt♦♥ s r♣ ♣ès Gr r♣ ♠♥s Gw

P♦r étr♠♥r s♦♥ trt ♥t étr♠♥ t♦t ♦r ♣è ♥s q s tr♦ ♣s ♣è ♦ù s♦t s r♥r s ① ♣ès ♦rrs♣♦♥♥t à ①♥÷s Gr trt st ♦rs ♦t♥ ♥ ts♥t ♥ ♦rt♠ ♣s ♦rt ♠♥ t②♣ str ❬❪ ♥tr s ① ♥÷s ♦rt♠ rt♦r♥ ♥ ♥s♠ ♥÷s Gw ♦♥stt♥t s ♦ts ♥tr♠érs ♣♦r tt♥r s st♥t♦♥♦s s ♦ts ♥tr♠érs s♦♥t ♣r♦rs tért♠♥t ♣r ♥t à ♠♦è ♦rs rts sé sr s str♥ ♦rs

♦té ♣tt é sé sr s str♥ ♦rs

♣r♥♣ str♥ ♦rs été ♥tr♦t ♥s ❬❪ ♥ érr s ♥trt♦♥s ♥ts r ♥r♦♥♥♠♥t s ♥ts s♦♥t ♦rs ss♠és à rs♥trs ♠sss t ♦♥ ♥q♠♥t ♥ ttrt ♣♦s ♣r♥♣ st ♠ttr ♥♠♦♠♥t q ♥t ♥ ♣♣q♥t ér♥ts ♦rs sr rs ♥trs ♠sssrs♣ts ♥s ♣♦r ♥ ♥t s ér♥ts ♦rs ♣♣qés à s♦♥ ♥tr ♠sss♦♥t

❯♥ ♦r q ttr ♥t rs s♦♥ ♣r♦♥ ♦t ♥tr♠ér étr♠♥é ♥st♦♥

s ♦rs ré♣s♦♥s és à ♥r♦♥♥♠♥t q é♦♥ ♥t s ♠rs s ♦rs ré♣s♦♥s és ① trs ♥ts ♣♦r étr q ① ♥tss♥tr♦s

rést♥t s ♦rs st ♥ ♦r q é♣ ♥t rs s♦♥ ♦t t♦t♥ ét♥t s ér♥ts ♦sts q ♣s t♠♣s ♠é♥q s rs sér♥ts ♦rs t ♦♥ rést♥t s♦♥t ♠ss à ♦r ♥ ssrr ♥ ♠♦♠♥trést


♦r♥st♦♥ rés ♥ r♣

♥ ♣s ♠♦té st ♥éssr s♠r s ♦♥♥ttés ♣♦t♥ts ♥trs ér♥ts ♥ts P♦r r ♥ r♣ rés① Gn st ♥tr♦t

Gn st ♥ ♠tr♣ ♦r♥té ❯♥ ♠tr♣ ♥q q ① ♥÷s ♣♥têtr rés ♣r ♣s ♥ ♥q ♥ s ♠t♣s ♥s ♣r♠tt♥t ♦t♥r s♥♦r♠t♦♥s P ér♥ts ♥ ♦♥t♦♥ s tst♦♥ ♥ r♣ ♦r♥té♣r♠t r♥r ♦♠♣t ss②♠étr s♦♥ q ♣t ♣♣rîtr ♦rsq♦♥♦♥sèr ♥ ♣ss♥ é♠ss♦♥ t♦ ♥ s♥sté ré♣t♦♥ ér♥t ♣♦r♥ s tr♠♥① s ♥÷s r♣ rés ♦♥t♥♥♥t s ♥♦r♠t♦♥ss♣é♥t rs s rs ♣ss♥s é♠ss♦♥ ♦ rs s♥stés r♦ r s♥s Gn s♦♥t st♦és s rs Ps é♥és ♥tr s ér♥ts ♥ts srs s Ps s♦♥t ♣♦r ♠♦♠♥t és à ♠♦è ♠t ért♥s t ♣♦rr♦♥t êtr ♣r♦♥♠♥t ♦t

♥ ♣♦♦r rssr s ♣r♦è♠s ♦st♦♥s strés ♦♥♣t rés① ♣rs♦♥♥s st ♥tr♦t ❯♥ rés ♣rs♦♥♥ st ♥ r♣ st♦é ♥s ♥♥÷ r♣ Gn r♣rés♥t s♦♥ qà ♥ ♥÷ rés ♥s ♣t êtr ♦♠♠ ♥ rstrt♦♥ r♣ Gn ♥tré sr ♥ ♥÷ k t ♠té ① ♦s♥s k♣rt♥t s ♠ê♠s s r str ♥ rés Gn ♥s q s rés①♣rs♦♥♥s ss♦és à ♥ s ♥÷s

tt sr♣t♦♥ s rés① ♣rs♦♥♥s ♣r♠t rér ♥ s♣ résré ♣♦rq ♥÷ ♦ù st ♣♦ss ♠îtrsr s ♥♦r♠t♦♥s ♦♥t ♥÷ s♣♦s ts ♥♦r♠t♦♥s q ♥÷ ♣t rr ③ ss ♦s♥s r♥r ♣♦♥t st éré♣r ét ♦♠♠♥t♦♥ ♥tr ♥÷s

é♥s♠ ♦♠♠♥t♦♥ ♥tr s ♥ts

q ♥t s♣♦s ♦s ♦♠♠♥t♦♥ érés ♣r ♠é♥s♠s st♥ts ❯♥ ♠é♥s♠ ré♣t♦♥ q tt♥ q♥ rqêt ♣r♥♥ ❯♥ ♠é♥s♠ é♠ss♦♥ ♦♠♠♥é ♣r ét ♦st♦♥ ♥ ♥♦②r s rqêts♦rsq♥ ♥t r à s ♣♦st♦♥♥r t q st ♥ r♥ ♥♦r♠t♦♥ ét ♦st♦♥ ♥♦ ♥ rqêt à ♣rt tr♥s♠ss♦♥ ét ♦♠♠♥t♦♥ r♥r ♥trr♦ ♦rs rés ♣rs♦♥♥ ♥÷ ♣♦r ♦♥♥îtr s ♥÷s qs♦♥t ♥ stés ♥ ♦♥t♦♥ ♦♥rt♦♥ é♥ ♣r tstr ♥ rqêtst ♦rs ♥♦②é à t♦t ♦ ♣rt s ♥÷s rés ♣rs♦♥♥ ♣rt ré♣t♦♥ ♥ s ♥÷s ♦♥r♥és ♠t à ♦r rés ♣rs♦♥♥ ♥÷ ②♥t té ♠♥ t②♣ ♥♦r♠t♦♥s ♠ss à ♦r s♦♥t ♦♥rés ♣r tstr s♠tr ♣r♠ ♥tr trs s ér♥ts Ps t♦ ♥ ♣♦st♦♥ st♠é

①♠♣ tst♦♥ s♠tr ②♥♠q

tt ♣rt é♠♦♥tr tté ♣t♦r♠ ②♥♠q trrs ①♠♣s ♣r♠r s♥térss à ♣r♦t♦♥ ②♥♠q Ps t s♦♥ à ♦st♦♥ stré


r♣ Gn t s rés① ♣rs♦♥♥s ss♦és à q ♥÷


t♥t♦♥ s Ps sr ♥ trt♦r s②♥tétq

♥ ♦♥sèr sé♥r♦ stré ♥ r ♦ù ♥ ♥t s é♣ ♥s ♥♥r♦♥♥♠♥t t st ♥ ♦♥♥①♦♥ ♣♦t♥t ♣♦♥ts ès r♣rés♥tés ♣rs tr♥s trt♦r ♥t st r♣rés♥té ♣r s ér♥ts ♣♦♥ts r♦sq ♣♦♥t r♦ ♦rrs♣♦♥ à ♥ é♥t♦♥ ♣♦st♦♥ à ♥ ♥st♥t t t st sé♣ré 1s s♦♥ ♦s♥ s ♣♦♥ts r♦s ♥♥t s ♥♦♠rs s♦♥t s r♣èrs ♣r♠tt♥t s rr rt♦♥ é♣♠♥t

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rt♦r ♥ ♥t ♠♦ ♥ ♣rés♥ ♣♦♥ts ès sttqs

r r♣rés♥t s ér♥ts rs ♦ t és à ♠♦è ♠t ♣♦r s ér♥ts ♣♦st♦♥s é♥t♦♥♥és ♥t s r♣qs r♣rés♥t♥t s ér♥ts ♥s ♥tr ♥t ♠♦ t s ♣♦♥ts ès

t♥t♦♥ à ♣rtr ♥ trt♦r s②♥tétq

♠♣é♠♥tt♦♥ ♦t ♥ ♣r♠tt♥t ♣s ♥♦r s♠t♦♥ t♠♣sré s ♦♥♥és s♥ts s♦♥t ♣♦sttrtés r ♣rt r ♦♥ ♦sr s r②♦♥s ♦t♥s ♣♦r ♣♦st♦♥s ér♥ts r ♣rtr♦t s♦♥t és s s ♦rrs♣♦♥♥ts ♥t s é♣ç♥t ♥s ♠ê♠ ♣ès s♥trs ♣r♠tt♥t ♦t♥r s r②♦♥s ♣r♠èr s♦♥t rétsés ♣♦rr s trs

é♠♦♥strt♦♥ ♦♠♠♥t♦♥ ♥tr ♥ts

P♦r é♠♦♥trr ♥térêt s♠t♦♥ ♦♠♠♥t♦♥ ♥tr ♥ts ♦♥s♥térss sé♥r♦ stré ♣r r ♦ù ♥ts s r♦s♥t ♥s ♥ ♦♦r♥ s ♥ts ♦♥t s ér♥ts ♣♦st♦♥s s♦♥t r♣rés♥tés ♣r s ♣♦♥ts r♦st rts ♣♥t ♦♠♠♥qr sr s t ♥♦♥t q♥ ♥ ♦♠♠♥ rs♦♥ ♥ r ♣r♠t ♦♠♠♥qr q ♥rs r♣rés♥tr ♣r s ①tr♥s r♦s ♣♦r ♣r♠r ♥t t ♣r s ① tr♥s rts ♣♦r s♦♥♥t ♥ s ♥÷s ss② s ♣♦st♦♥♥r à q rrîss♠♥t ♣♦st♦♥


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♦t♦♥ s rs ♣ss♥ rç t ♦ s♠és ♥tr ♥t♠♦ ♥rs sttqs ét♦♥ s Ps st té ♥ t♠♣s ré râ ♠♦è ♠t


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①♠♣ ♦t♥t♦♥ ♣♦r ♣♦st♦♥s ♥t ér♥ts


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♥r♦ s♠t♦♥ ♣♦r strr ♥térêt ♦♠♠♥t♦♥ ♥tr♥ts ♥t 1 r♣rés♥té ♣r r♦♥ r♦ ♥ ♣t s ♦♥♥tr q① ♥rsr♣rés♥tés ♣r s tr♥s r♦s ♥t 2 r♣rés♥té ♣r r♦♥ r♦ ♥ ♣t s♦♥♥tr q① ♥rs r♣rés♥tés ♣r s tr♥s rts s ① ♥ts ♣♥té♠♥t ♦♠♠♥qr ♥tr ① ss s♦♥t ♥ sté

rrr ♣♦st♦♥ s ♥ts ♥ ♦♥t♦♥ t♠♣s s♠t♦♥ st ♣rés♥té♥ r ♣♣rît q rrr ♣♦st♦♥♥♠♥t r♦t ♣♦r s ① ♥tssqà s ♦ s ♣s ér♦t rt♠♥t à t ♥st♥t t ♥st♥t ♥qé ♣r r♣èr ♥♠ér♦ 1 ♣♦r ♥t r♦ ♦rrs♣♦♥s ♠♦♠♥t ♦ù s ① ♥ts s♦♥t ♥stés s s s♦♥t ♦rs é♥é rs st♠t♦♥s ♣♦st♦♥s t ♥ ♦ q r ♣r♠s s ♣♦st♦♥♥r ès ♦rs qs sé♦♥♥t ♥ tr s ♥ s♦♥t ♣s ♥sté t ♥ ♣♥t ♦♥ ♣s é♥r ♥♦r♠t♦♥ rrr ♣♦st♦♥ r♦t ♥♦

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rrr st♠t♦♥ ♣♦st♦♥ ♣♦r s ♥ts ♥ ♦♥t♦♥ t♠♣s

t♦♥ s ♠ét♦s ♣r♦♣♦sés à ♦♥♥és ♠srés

♦♥s♦♥

tt ♣rt sst ♦sé sr ér♥ts s♣ts ♣t♦r♠ s♠t♦♥②♥♠q ♦t ♦r ♠♦té s ♥ts été ♦ré ♥ ♣r♦♣♦s♥t ♥sr♣t♦♥ ♠♦té s ♥ts sé à ♦s sr s r♣s t s str♥♦r ♥st ♠♦ést♦♥ ♦r♥st♦♥ s rés① été ♣rés♥té à r♣ rés ♦t♠♠♥t ♥♦t♦♥ r♣ ♣rs♦♥♥ s♠♥t s♦♥q♥ ♥÷ ♣t ♦r rés été ♥tr♦t ♣♦r ♣r♠ttr s s♠t♦♥s sé♥r♦s strés ♥térêt tt ♦♥t♦♥♥té st r♥♦ré ♣r ♥tr♦t♦♥♥ ♠é♥s♠ ♦♠♠♥t♦♥ ♥tr s ♥÷s ♣r♠tt♥t s♠r s é♥s♥♦r♠t♦♥s ts q s Ps ♥♥ ♥ r♥èr ♣rt ♣r♦♣♦sé ① ①♠♣stst♦♥ ♣t♦r♠ ②♥♠q ❯♥ ♣r♠r ①♠♣ ♦ù s Ps ♥ ♥ts♦♥t és ♦rs s♠t♦♥ t ♦ù s s♦♥t éés ♥ ♣♦sttrt♠♥t❯♥ s♦♥ s♠t♦♥ stré ♥ ♠é♥s♠ ♣♦st♦♥♥♠♥t ♦♦♣ért r♥♣♦ss ♣r ét ♦♠♠♥t♦♥

t♦♥ s ♠ét♦s ♣r♦♣♦sés à ♦♥♥és♠srés

♣tr ♣r♦♣♦s ét♦♥ s ér♥ts ♦ts érts ♣réé♠♠♥t ♥ ts♥t s ♦♥♥és rés ♦t♥s ♣r ♥ ♠♣♥ ♠sr ♠♥é ♦rs ♣r♦tr♦♣é♥ P ❲ ❬❪

①♣♦tt♦♥ s ♦♥♥és ♠sr ❲

♠♣♥ ♠sr q été résé ♥s r ♣r♦t r♦♣é♥ P❲ st ♣r♠èr ♠♣♥ ♠sr ♣q ♦st♦♥ étér♦è♥♥♥t s t♥♦♦s tr r ♥s ❬❪ Pr♠ s ér♥ts sé♥r♦s résés ♦rs tt ♠♣♥ ♦♥t été rt♥s ♥ ♣♦r ①trr s ♣r♠ètrs s♠♦ès Ps tr ♣♦r ér s ♦rt♠s

①trt♦♥ s ♣r♠ètrs s♠t♦♥

t rés♠ s ér♥ts rs s ♣r♠ètrs ♠♦è ♣t♦sss♦♥ ♣♦r s P ♣ss♥ ♠ê♠ ss♠♥t ♥ ♠♦è ss♥ ♣♦rrrr s ♦s s ♣r♠ètrs ①trts s♦♥t ♦♥♥és ♥s t

♦♠♣rs♦♥ s ♠srs s♠t♦♥ ♠t

s rs t ♦♠♣r♥t s rs s t ♦ ♣♦r t♦s s ♣♦♥ts♠srés ♦rs ♠♣♥ ♠sr ① rs ♦t♥s à s♠t♦♥♠t ❯♥ ♦♥♥ ♦rrs♣♦♥♥ s♠t♦♥ ♠srs ♣t ② êtr ♦sré


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♦♠♣rs♦♥ ♠srs s♠t♦♥s ♣ss♥ rç ♣♦r ♥rs ♦r♥ss♥t s ♦srs ♥ ♦♥t♦♥ s ér♥ts ♣♦♥ts ♠srs s ♣♦♥tss r♣rés♥t♥t s ♦♥♥és ♠srés s ♣♦♥ts ♥♦rs r♣rés♥t♥t s ♦♥♥éss♠és ♠t t s r♦① r♦s r♣rés♥t♥t rrr


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♦♠♣rs♦♥ ♠sr s♠t♦♥ ♣ss♥ rç ♣♦r ♥rs ♦r♥ss♥t s ♦srs ♦ ♥ ♦♥t♦♥ s ér♥ts ♣♦♥ts ♠srs s ♣♦♥tss r♣rés♥t♥t s ♦♥♥és ♠srés s ♣♦♥ts ♥♦rs r♣rés♥t♥t s ♦♥♥éss♠és ♠t t s r♦① r♦s r♣rés♥t♥t rrr


Pr♠ètrs ♠♦è ♣t♦ss ①trt s♥r♦

♠ér♦ ♥÷ RSS0 dB np σ dB

♦s

Pr♠ètrs ♠♦è rrr st♥ ❯❲ ①trt s♥r♦

♥r♦ rrr st♥ m r♥ m

♦♠♣rs♦♥ ♠srés t ♣r♦ts ♦t

r ♦♠♣r s s ♠srés t s♠és ♣r ♦t ♦♠♣rs♦♥ ♠♦♥tr q ♦t ♣r♠t rtr♦r ♣r♠r trt ♥s q s♦♥trtrs ♣r♥♣① ♥ ♦sr ss q rt♥s trts rtr♦és ♣r s♠t♦♥ s♦♥t s♥ts ♠sr t t st ♣r♦♠♥t û à s ♦tt♦♥sés à ♥ ♦st ♥♦♥ r♣rés♥té ♥s ♥r♦♥♥♠♥t s♠t♦♥

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linearlevel(V)

Simulation

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UWB CEA Measure

♦♠♣rs♦♥ ♥ ♦t♥ ♣r ♠sr t s♠é ♦t rtr ré 21.8 ns t r ♠sré st 22.17 ns ♥ r♦ r♣rés♥t rtr ré


t♦♥ s ♣r♦r♠♥s ♦rt♠ P

ét♦♥ P sr s ♦♥♥és ♠srés ♠♦♥tr q ♠ét♦ ♣rétsr ♦rsq s♦♥t tsés s ♦srs ♦s ss ♦ ss♦és à s ♦srs ♣ss♥s ♠ét♦ ♦r♥t ♠♦♥s ♦♥s réstts q ♦rsq♦♥ts ♥q♠♥t s ♦srs ♣ss♥s ♦♠♣♦rt♠♥t st ♠♣t à ♦♠♣①té tst♦♥ s ♦srs ♠ét♦ P

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s rrr ♣♦st♦♥♥♠♥t ♦♠♣r♥t P

t♦♥ ♦rt♠ ♦st♦♥ tst ②♣♦tèss

♠ét♦ tst ②♣♦tès P t ♥ ♠ét♦ ♦♥t été ♣♣qéssr ♥ sé♥r♦ ♦♠♣t♥t ♦ t ♥ s ér♥ts rs t r♥s ♦t♥s s♦♥t r♣♦rtés ♥s t s réstts ♠♦♥tr♥t q ♠ét♦ tst ②♣♦tès ♣r♠t rér rrr ♠♦②♥♥ ♣♦st♦♥♥♠♥t♦t♦s r♥ rrr ♦t♥ ♠ét♦ ♣r♦♣♦sé st ♣s ♠♣♦rt♥tq♥ ts♥t ♥ ♣t s①♣qr ♣r ♥tr tst és♦♥ ❯♥rrr és♦♥ ♦rrs♣♦♥ à ♥ ♦① rr♦♥é ③♦♥ ♣r♠ s ① é♠tés


♣r s ♦s ♥s ♣♦st♦♥ ♦s ♣t êtr très é♦♥é ♣♦st♦♥ ré♥trî♥♥t ♥s ♥ rrr ♠♣♦rt♥t

rrr qrtq ♠♦②♥♥ ♣♦st♦♥♥♠♥t P t P

ét♦ rrr ♣♦st♦♥ ❬♠❪ r♥ rrr ♣♦st♦♥ ❬♠❪

P

P

♦♥s♦♥

♣tr ♣r♠s r s ér♥ts ♠ét♦s ♣r♦♣♦sés ♥s s ♣trs♣réé♥ts s ♦♥♥és sss ♥ ♠♣♥ ♠sr ♠t t ♦t ♦♥t ♠♦♥tré rs ♣tés à ♣rér s rs Ps t à rtr♦r s♣r♥♣① ♦♥trtrs s ♦♥♥s ♣r♦r♠♥s ♣♦st♦♥♥♠♥t ♦rt♠ P ♦srés ♥ s♠t♦♥ ♦♥t été rtr♦és ♥ ts♥t s ♦♥♥és♠srés ♥♠♥t ♠ét♦ ♦st♦♥ sé sr tst ②♣♦tèss ♠♦♥tréq ♣♦t rér rrr qrtq ♠♦②♥♥ ♣♦st♦♥♥♠♥t ♦♠♣ré à ♥♣♣r♦ t②♣

♦♥s♦♥ é♥ér

tt tès t ♣♦r t rssr ♣r♦è♠ s♠t♦♥ ②♥♠q ♣r♦♣t♦♥ ♦♥s étr♦♠♥étqs ♥ ♥térr ♥♦t♠♠♥t ♣♦r s ♣r♦é♠tqs ♦st♦♥ r♦s ♣r♥♣① s♣ts ♦♥t été ♦rés ♣r♦♣t♦♥ s♦♥s s ♦rt♠s ♦st♦♥ t é♥t♦♥ ♥ ♣t♦r♠ s♠t♦♥②♥♠q ♣tr rés♠ s ♣r♥♣① réstts ♦t♥s t s ♦♣tqs rrs trs

♦♥s♦♥s

♦♥♣t s♥tr r②♦♥ été ♥tr♦t ♣♦r érr t ♣rsst♥ ♠♥ ♦sr ♥s s ♦rs éèrs rt♦♥s ♣♦st♦♥s é♠ttr ♦ ré♣tr s♥tr é♥ ♦♠♠ ♥ séq♥♥trt♦♥s t ♠tér① ss♦és st s♣éq à t♦♣♦♦ ♥r♦♥♥♠♥t

❯♥ sr♣t♦♥ s♦s ♦r♠ r♣ ♦t ♣r♠s ♠♦ésr♥r♦♥♥♠♥t ♠s é♠♥t s rt♦♥s ♥tr s ♥trt♦♥s étr♦♠♥étqs ❯♥ t r♣rés♥tt♦♥ té tst♦♥ s s♥trs t sr♣t♦♥ ♠♦té

éért♦♥ ♥♥ré ♣r ♦r♠s♠ t♦rsé ♦t ♣r♠s résr s s♠t♦♥s tr r ♥ ♥ ♠♦té ♦r♠s♠ r♣♦s sr♥ sr♣t♦♥ ♥

♦♥s♦♥ é♥ér

♦t ♠t é♠♥t ♣r♦té tt t♦rst♦♥ t ♣r♠s ♦t♥rr♣♠♥t s ♦srs r♦s s♠és ♣t♥t ♦♠♣rs♦♥ ①♦t♥s ♥ ♠srs

♠ét♦ ♦st♦♥ étér♦è♥ P ♠♦♥tré très ♦♥♥s ♣r♦r♠♥s ♥ s♠t♦♥ t sr s ♦♥♥és ♠srés ♦♠♣rt♠♥t à ♥ ♦rt♠

❯♥ ♠ét♦ sé sr ♥ tst ②♣♦tèss ts♥t s ♦srs ♣ss♥♣♦r ér ♥tr ① ♣♦st♦♥s st♠és ♦♥rr♥ts été ♣r♦♣♦sé ♦♠♣réà ♥ tst♦♥ ♠ét♦ ♣r♠s rér rrr qrtq♠♦②♥♥ ♣♦st♦♥♥♠♥t sr s ♦♥♥és ♠srés ♦rsq ♣ ♦srs ♣ss♥s s♦♥t s♣♦♥s

r♦s s♣ts ♣t♦r♠ ②♥♠q ♦♥t été ♥tr♦ts ♠♦té ♦r♥st♦♥ rés t ♦♠♠♥t♦♥ ♥tr ♥♦s ♣r♠tt♥t s♠t♦♥ sé♥r♦s②♥♠qs étér♦è♥s t ♦♦♣érts

tr tt tès srt s♣♣♦rt à ♥ ♣r♦t ♦♣♥ s♦r P②②rs ❬❪♥s s ér♥ts ♦ts ♠t ♣t♦r♠ ②♥♠q t s ♠ét♦s ♦st♦♥ ♣rés♥tés ♥s s ér♥ts st♦♥s s♦♥t téérs ttss rtt♠♥t sr st

①s rr tr

P♦r ér à ♥ r②tr♥ ♥ré♠♥t ♥ ♦rt♠ ♣r♠tt♥t ♠s à♦r s s♥trs srt ♥éssr ♥ t ♥ s ♠♦♠♥t rt♥ss♥trs s♦♥t ♠♥és à s♣rîtr ❯♥ ♣r♦éé ♣r♠tt♥t r♠♣r ss♥trs s♣rs ♣r ♥♦s rst à étr♠♥r

rs♦♥ t ♦t r②tr♥ t♦rsé ♥ ♣r♥ ♣s ♥ ♦♠♣tét♦♥ s ♥trt♦♥s t②♣ rt♦♥ ♦t♦s t②♣ ♥trt♦♥rt êtr très ♣r♦♥♠♥t ♦té à ét♦♥ ♥ ♣sq s ér♥ts éts s♠t♦♥ s♦♥t éà ♦♠♣ts

❯♥ s ♠é♦rt♦♥s r♥s ♣♦sss ♣r t♦rst♦♥ ♦t r②tr♥ st ♣♦ssté tsr s t♥qs stré ♣♦r ér sér♥ts r②♦♥s tt ♠é♦rt♦♥ ♠értrt êtr ♠♣é♠♥té ♥ rér♥ ♥♦ ♦s t♠♣s trt♠♥t ♦t

P ♣♦rrt êtr ♥sé ♥s s sé♥r♦s ♦st♦♥ ♦♦♣érts tstrés P♦r ♥tr s ♦♥♥és é♥és rst à étr♠♥r

♦rt♠ tst ②♣♦tèss ♣♦rrt êtr é♥érsé à s stt♦♥s ♠t♠♦s ♠♣q♥t ♣s ① ③♦♥s t é♠♥t êtr ①♣r♠é ♣♦r ér♥ts♠♦ès rrrs ♣ss♥s rçs

♦♥r♥♥t ♣t♦r♠ ②♥♠q r♣rés♥tt♦♥ s ♥ts ♣♦rrt é♦rrs ♥ r♣rés♥tt♦♥ ♦r♣♦r ♥ rssr s ♣r♦é♠tqs rés① ❯♥ t r♣rés♥tt♦♥ ♣r♠ttrt ét ♣♦ssé ♥♥ ♦r♣s♠♥ ♥s ♥

♣r♦t♦♦ ♦♠♠♥t♦♥ sq ♣r♦♣♦sé ♣♦rrt s♥rr ♥♦t♠♠♥t♥ ♠♣é♠♥t♥t s ♠é♥s♠s ♦ ♥ ♣rtr t ét ♦♠♠♥t♦♥ ♣♦rrt ♥térr ♦t s♠t♦♥ ❲ ❬❪

♣♣♥①

t ♠♥s♦♥ rr②

♥ s s ♥ ①t♥s♦♥ ♦ t ♠tr① ♥ ♠♥s♦♥s s ♦♥♣t ♦ ss ♦♥♥♥t t♦ sr r ♦♣rt♦♥s ♥♦♥ r ♥♠r ♦ ♠♥s♦♥srt♦♥ ♠t♠t ♥♦tt♦♥s r② ♦ t♦ ①♣rss t s ♥ ♠♣♠♥t♦♠♣tt♦♥② ♦r tt rs♦♥ ts ♣♣♥① ♥tr♦s r♦s ♥♦tt♦♥ ♥ ♦♣rt♦♥s ♦♥♥t♦♥ rrr♥ t♦ s tr♦♦t ts ♦r

s t strtr s r② t strtr t s srt s ♠♥s♦♥sr ♥♠ ①s ♥ ♥t ♥t ♥ t ♦♦♥ t ♦r ①s s t♥ s s②♥♦♥②♠♦s ♦ ♠♥s♦♥ ♥ ♠♥s♦♥ s ss♦t t ♥ ①s ♥ ♥ s♦ s♦♥ s ♥♠♣②rr② strtr ❬❪ ss ♠♣② s ♣②t♦♥rr② ♣r♦s ♠t♦s ♦r ♠♥♣t♥ s r② ♦s r♦♠ ♠t♠tt♥s♦r s ♠♣② s r② ♦♥s ♥ ♥t ♦r ♥ t s strt♦ts r t ♦♠tr ♥tt♦♥ s ❲♥ ♦♠s t t♠ t♦ rt ♠t♠t② t s ♦ t ♠♣② sr♣rs♥② ♦sr tt t t♥s♦rs ♦♥♥t♦♥ s ♥st♥ s♠ ♦r ①♠♣ s ♥♦t t♦ ①♣rss s ♦♣rt♦♥s r♦st♥ ♣r♣♦s ♦ t ♦♦♥ s t♦ ♣r♦ ♦♥s ♥♠♦s♠t♠t st ♦ ♥♦tt♦♥s t♦ ①♣rss t s ♥ t② ♦♠♣tt♦♥②♠♣♠♥t

♥r ♥♦tt♦♥ ♦♥♥t♦♥

♥ t ♦♦♥ ♥♦tt♦♥ ♥s♣r ② t ♥st♥ ♥♦tt♦♥ ❬❪ s ♥tr♦ ♥♥st♥ ♥♦tt♦♥ t s♠ ♦♣rt♦r s ♦♠tt ♥ ♦t s♣rsr♣t ♥ ♦r sr♣t rs t♦ ♥t ♥s ♦ ♦♦r♥ts t♦ ♠t♣ s♣rsr♣t ♥s rrt♦ ♦♠♥ ♥①s rs t ♦r sr♣t ♥s rr t♦ r♦ ♥①s s ♥♦tt♦♥s r② ♦♥♥♥t ♥ ♣r♦r♠t t ♦r ♦♣rt♦♥s ♥ ♠♥s♦♥ t ♦ ♦♥♥♥t t♦ ♣ tr ♦♥ t ♥♦ ①s r♥ t ♠t♣t♦♥ ♦r♦rtt ♥♦tt♦♥ ♥tr♦s ♥♦♥②♠♦s ♥①s s i, j, k, ... ♥ ♥rs t r♥♦♠♣①t② ♥ s ♦ ♠♥s♦♥s

♦r tt ♣r♣♦s ♣r♦♣♦s ♥ ♦t♦♥ ♦ t ♥♦tt♦♥ ♥ ♦r♥ t t♦♦♥ rs

♣♣♥① t ♠♥s♦♥ rr②

t s♠ ♦♣rt♦r s ♠♣t t rt ♦r sr♣ts ♥t t ①s ♥♠s ♦r ♥ s ♦ ♦♣rt♦♥ t ①s

♥♦t rt② ♥♦ ♥ t s t rt s♣rsr♣ts r s t♦ s♣② t ①s rt② ♥♦ ♥ ♦♣rt♦♥ ♥r♥ ①s ♦ t rt s♣rsr♣t ♥t t ①s t♦ ♠t♣ t♦tr

♥ r♣♠♥t ♦ ♥st♥ ♥♦♥②♠♦s rs t ①s r ♥ t ①s ♥♠ sr t s♠ r ♥♠ ♥ ①♣t②

♥t t ♥tr ♦ t ①s t s♣ ♦ ♥ ①s ♥♠r ♦ ♠♥ts ♥ t ♥ ①s ♦s ♥♦t ♣♣r ♥

t ♥♦tt♦♥ ♥ s t♦ ♣♦♥t ♦t sr t s♣ ♦ s t♣ ♦ ♥trs ♦♥t♥s t ♥t ♦ ①s t t s♣rsr♣t s ♦♣t♦♥② s t♦ sr♠♥t s t t ♦r sr♣t s ♦♣t♦♥② s t♦ s♣② ♣♥♥t ♣r♠trs ♦ t

s♣t rt ♦r sr♣ts ♥t t ①s ♥♠s t s ♣♦ss t♦ rss s♣

uth ♠♥t ♦♥ ①s j ♦r ♠♥t ♦♥ ①s i ♦ Ai,j ①s t ts ♥♦tt♦♥

Ai,j[u]

♣rt♦♥ rs

ss② t ♦t ♣r♦t C ♦ ♠trs A ♥ B t ♠♥s♦♥ (I, J) ♥(J,K) sr♥ ♦♠♠♦♥ ①s ♦ ♥t J ♥ rtt♥ s t s♠♣ s♠ ♦ ♣r♦ts

ci,k =

J−1∑

j=0

ai,jbj,k

r ai,j ♥ bj,k r t ♠♥ts ♦ A ♥ B ♥ ci,k t ♠♥ts ♦ t rst♥♠tr① ♣r♦t C

♥ r ♠♥s♦♥s ♠trs r r♣ ② s ♥ ts s♠♣ ♦♣rt♦♥♥ r♣r♦ ♥r t ♦♥t♦♥ ♦ ♦♥♦r♠t② ❬❪ ♦♥♦r♠t② ♠♥stt s ♥♦ ♥t♦ s ♥ ♦♥② ①s t r♥t ♠♥s♦♥ ♥ A ♥ B r t♦ ①s s t ♠♥s♦♥s (I, V,K,L,M) ♥(V, J,K,L,M) rs♣t② t s ♣♦ss t♦ rt ♣rtr ♦t ♣r♦t rt② t♦♦♠♣t ①s v ♥ ♣♦st♦♥ ♥ rs♣t②

ci,j,k,l,m =V−1∑

v=0

ai,v,k,l,mbv,j,k,l,m

r ai,v,k,l,m ♥ bv,j,k,l,m r t ♠♥ts ♦ A ♥ B rs♣t② ♥ ci,k t♠♥ts ♦ t rst♥ ♣r♦t C ❯♥♦rt♥t② ♥ r ♥♠r ♦♦♠♣t ♠♥s♦♥s ①sts t s ♣♦ss t♦ ♠♥♥ t♦ sr r♥t ♦t♣r♦ts ♥ t ♦♠s ♥ssr② t♦ ♥ ② t♦ ♠ ①♣t ♦♥ ①s ts♠♠t♦♥ s ♦♥ t♦ ♣♣

♣ ♦tt♦♥s ♥ ♣rt♦rs

❯s♥ t ♦♥♥t♦♥ rs sr ♥ qt♦♥ ♥ rtt♥

Ci,j,k,l,m = Ai,vk,l,mB

v,jk,l,m

♥ t ①s ♦rr ♦ A s ♥♦♥ t s ♣♦ss t♦ s♠♣② t ♥♦tt♦♥ ②♦♠tt♥ t ①s ♥♦t ♥♦ ♥ t s ♥ ♦♠s

Ci,j,k,l,m = Ai,vBv,j

s t ♦t ♣r♦t ♥ r s ss ♦t ♣r♦t t♥ ♠trs♦ ♠♥s♦♥ (I, V ) ♥ (V, J) r t ♦♣rt♦♥ ♦ ♣r♦♣t ♦♥ tr♠♥♥ ①s k l ♥ m ❯s♥ ts ♥♦tt♦♥ t ♦t ♣r♦t ♦ ♠tr① ♥ tstr♥s♣♦s ♥ ♣r♦r♠ ♦♥ ② s♠♣② s♠♠♥ ♦♥ t ♦♠♠♦♥ ①s

Ak,l,mATk,l,m = A

i,vk,l,mA

i,vk,l,m

ATk,l,mAk,l,m = A

i,vk,l,mA

i,vk,l,m

♥ s ♦ ♦♠♥ ♦r r♦ t♦r t tr♥s♣♦st♦♥ s ♥♦t ② ①♣t♥ ♥♥tr② ①s s♣ ♦ ♦♠♥ t♦r Ai s (i, 1) ♦tr ♣r♦t ♦ Ai

Ci,i,k,l,m = Ai,1k,l,m A

i,1k,l,m

♥♥r ♣r♦t sr ♣r♦t s

Ck,l,m = Ai,1k,l,m A

i,1k,l,m


r♦st♥

♥♦tr s♣t ♦ s rt② r r♦♠ ♠♣② P②t♦♥s rr② ❬❪ ❬❪ st r♦st♥ r♦st♥ ♦s t♦ ♦♠♣t s ♦ sr t st ♦♥ ①st t s♠ s♣

s s♦♥ ♦♥ r t ♠ss♥ ①s r t r♦♠ t r♣tt♦♥ ♦ ①st♥①s t♦ ♦ ♦♣rt♦♥ ♦♥sr♥ Ai,j,k,l ♥ Bj,k,l t s ♣♦ss t♦ ♣r♦r♠ t ♦t♣r♦t ♦♥ ①s k ♥ l t t s♠ t♠ r♦st♥ ①s i t

Ci,j,k,l = Ak,li,j B

k,l⋆,j

♣♣r sr♣t s t ♣ r s♠♠t♦♥ s ♦♥ s ♥♦tt♦♥ ♦s t♦ ♥♦ts ♥♦t t s♠ ♥♠r ♦ ①s

⋆ s②♠♦ s s t♦ ♥t ①s s ♥ r♦st

♣♣♥① t ♠♥s♦♥ rr②

r♦st♥ ♦ s t s♣ (I, J) ♥ (1,K) rs♣t② ♦♠♥s ♦ t s♦♥ r r♣t t♦ t ①s j t♦ t t t rst ♦♥♠t♣t♦♥ s♣ ♦ t rst♥ s (I, J,K)

♦♥t♥t

s ♥tr♦ t ♦♥t♥t♦♥ ♦♣rt♦r ♦♥ ♥ ①s u|u| ♦♥t♥t♦♥

s t ♦♣rt♦♥ ♦♥sst♥ ♥ st♥ s ♦♥ ♥ ①s ♠♥s♦♥ ♦ trst♥ ♦♥ ts ①s s t s♠ ♦ t t♦ ♦trs

Ci,k,l,m = Ai,k,l,m

|

l|Bi,k,l,m

♥ t ♠♥s♦♥ ♦ ①s l ♦ ♦t Ai,k,l,m ♥ Bi,k,l,m s dim(l) = L t rst♥♠♥s♦♥ ♦♥ ①s l ♦ Ci,k,l,m s dim(l) = 2L ② ①t♥s♦♥ t s ♣♦ss t♦ rt

Ci,k,l,m =|

l|

M−1

0

Ai,k,l,m

r t rst♥ ♠♥s♦♥ ♦♥ ①s l ♦ Ci,k,l,m s dim(l) =M × L

❯♥♦♥

♥♦♥ ♦♣rt♦r⋃

♦s t♦ tr ♠♥ts ♦ t♦tr ♦r ♥st♥

Ci,k,l,m =L−1⋃

u=0

Ci,k,l[u],m

dim(l) = L


s♣

rs♣♥ ♦♣rt♦r ♦s ♥ t ♥♠r ♦ ①s ♦ ♥ ② r♦t♥s♦♠ ♠♥s♦♥s

Ci,k,lm = rs♣(i,k,lm)

(Ci,k,l,m)

r t ①s lm s③ s ♠(lm) = ♠(l)× ♠(m) ② ♦♥♥t♦♥ t rs♣ ①♥♠ s ♦♥t♥t♦♥ ♦ t ♦r♥ ①s

t♦♥ t♥ trs ♥ s ♦r ss ♦t Pr♦t

ts s♣♣♦s AB t ♠♥s♦♥ (I, J)(K,L) ❯♥r t ♦♥t♦♥ ttt ♥ ①s r ♦♠♣t t ♣♦t♥t r♥t ♣r♠tt♦♥s ♥ tr ss♦tP②t♦♥♠♣② q♥t r ♥ ②

ATB = Cj,l = Ai,jBk,l

BTA = Cl,j = Bk,lAi,j

⇐⇒ ♥♣♥s♠→

⇐⇒ ♥♣s♠❬♥①s❪❬♥①s❪①s

AB = Ci,l = Ai,jBk,l

BA = Ck,j = Bk,lAi,j

⇐⇒ ♥♣♥s♠→

⇐⇒ ♥♣s♠❬♥①s❪❬♥①s❪①s

ABT = Ci,k = Ai,jBk,l

BAT = Ck,i = Bk,lAi,j

⇐⇒ ♥♣♥s♠→

⇐⇒ ♥♣s♠❬♥①s❪❬♥①s❪①s

st ♦ rs

st ♦ rs

❯❲ t♦♠♠♥t♦♥ ♠ss♦♥ ♠s ♦r ❯ ♥ r♦♣ ♦ ♥t♦♥s sr♥ t ♣r♦♣t♦♥ ♥♥ ♥ ♦♥♥t♦♥ tr

t♥ ♥ ♣r♦♣t♦♥ ♣♥♦♠♥ ②♣ ♥♥ ♠♣s rs♣♦♥s ♦r t ♠t♣ strs ts ♥ ♦ ss ①♦♥ ♥ rrt♦♥ ss r♣ ♦ strtr Gs t♦♣♦♦ r♣ Gt r♣ ♦ ❱st② Gv 4 ♣♦t♥t t②♣s ♦ ♥trt♦♥s ♦♥ ♠tr ♥trt♦♥ r♣ Gi tr♠♥t♦♥ ♦ ♥tr♠t ♣♦♥ts tr♠♥t♦♥ ♦ ♥trt♦♥ ♣♦♥ts ♣rs♥tt♦♥s ♥ r♥t ①s ♦ s♠ r② ♥♥ t t P♦st♦♥s ♦ ♦♦r ♥trt♦♥s P♦♥ts ♦♦r t♦♥s strt♦♥ ♦ t s ♦ ♣r♠tr③t♦♥ ♥♣t ♣♥ ♥ ♦t♣t ♣♥ ①♠♣ ♦ t ♦r♥③t♦♥ ♦r r②s s♣♥ ♥ sst♦♥ ♦ s ♣rt♦♥s ♦♥ t t ♥trt♦♥ t♥♥ t ♣r♦♣t♦♥ ♥♥ ♦♠♣♥st♦♥ t ♦ ♦ rt♦♥ ♦♥ts ♦ ♦ tr♥s♠ss♦♥ ♦♥ts t♦♥ ♦♥ts ♦ ♠tr t ♦♠♣① ♣r♠tt②

①♠♣ ♦ ♦♦♣rt♦♥ ♠ t P ♦♥str♥ts ♦① t P ♦♥str♥ts r t P ♦①s ♦♥str♥ts qtr ♣♣r♦①♠t♦♥

st ♦ rs

s♦♥ ①♠♣ ♦♥ s♣t ♦① ♦♠tr ♦♥str♥t ♥ ♦① st♠t ♣♦st♦♥ s♥ P P s♥r♦ s ♦ ♣♦st♦♥♥ rr♦r s♥ P ♥ s♠t♦♥ P s♣ ♦♠♣tt♦♥ ♦♠♣rs♦♥ ♥r♦ ♦♥sr ♦t ②♣♦tss tst♥ ♠t♦ ♦♠♣rs♦♥ ♦ ②♣♦tss tst♥ ♠t♦ t♦♥ ♦ t ♥♥ ♦ t ♥♠r ♦ ♣♥ ♥♦ ss ♣ss♥ s♥r♦ ss ♣ss♥ s♥r♦ st♠t t P ss ♣ss♥ s♥r♦ st♠t t ②♣♦tss tst♥ ♠t♦ ♦ ♣♦st♦♥♥ rr♦r s♥ t ♣r♦♣♦s ②♣♦tss tst♥

♥ ①♠♣ ♦ r♣ ♦ r♦♦♠s Gr ♥ ①♠♣ ♦ r♣ ♦ ② ♣t Gw ss♦t t♦ t r♣ ♦ r♦♦♠ ♦r t♦r r♣ t♦r r♣ Gn ♥ t ss♦t ♣rs♦♥ ♥t♦rs ♦♠♠♥t♦♥ ♣r♦ss rt♦r② ♦ ♠♦ ♥t s♠t r♦ ♥s ♠t♦♥ ♦ s t♦♥ ♣r♦ss♥ t♠ ♥ ①♠♣ ♦ ♦♠♣tt♦♥ ♥ s s♦♥ t stt♦♥ ♦ t

s♠t♦♥ ♥ t r②s ♥rt ② t r② tr♥ ♦♠♣tt♦♥♥ t s♠t ♥ ♠sr s ♦ t stt♦♥ r r♣rs♥ts ♥t♦♥ ♦ ②

♠t♦♥ ♦ ♦♦♣rt s♥r♦ P♦st♦♥♥ rr♦r ♥ ♦♦♣rt s♥r♦

❲r ♠sr♠♥t ♣t♦r♠ ♥r♦ ♦ t ♠sr♠♥t ♠♣♥ sr♠♥t s♥r♦ sr♣t♦♥ ♥r♦ ♠sr ❩ r ♣♦r s ♥t♦♥

♦ t ♠sr ♣♦♥ts r♦♥ trt st♥ s ♥t♦♥ ♦ t♠sr ♣♦♥t r ♦ts ♦♥ t r ♣♦r ♦rrs♣♦♥ t♦ t♠♥ s ♦ r ♣♦r ♦r t ♥ ♠sr ♣♦st♦♥s

♦ ①trt♦♥ ①♠♣ ♦ ♣♦r ♠sr♠♥t t♥ stt ♥s t♥

♥ ♥ t♥ ♥ ♦♠♣rs♦♥ s♠t♠sr ♦♠♣rs♦♥ s♠t♠sr ♦

st ♦ rs

♦ ♣♦st♦♥ ♦♥rt♦♥s ♦ t r♦② ♥ t ♣r ♥♦ s ♦r ♦♠♣tt♦♥ ♥ ♦♥rt♦♥ s stt♦♥ ♥ ♦♥rt♦♥ s stt♦♥

r②s ♦t♥ r♦♠ t ♦♠♣tt♦♥ ♦r ♦♥rt♦♥ ♥ ♦♥rt♦♥

r②s ♦t♥ r♦♠ t ♦♠♣tt♦♥ ♥♥ ♠♣s rs♣♦♥ss ♥rt s♥ t ♣r♦♣♦s r② tr♥ t♦♦

♦♠♣r t♦ t ♥♦r♠③ ♠sr ♥♥ ♥r② ♣r♦ ♥ t r♦♥ trt ② s 32.3 ns rs t ② ♦rrs♣♦♥♥ t♦ t♠sr t♦ ② r♥♥ s 32.8 ns ♥ t r♦♥ trt ② s21.8 ns rs t ② ♦rrs♣♦♥♥ t♦ t ♠sr t♦ ② r♥♥s 22.17 ns r ♥ s♣rt♥ t r② ♣rt r♦♠ t t ♣rtr♣rs♥t t r♦♥ trt ②

♥♥ ♥r② ♣r♦ ♦ s♠t♦♥s ♥ ♠sr ♥ t r♦♥trt ② s 32.3 ns rs t ② ♦rrs♣♦♥♥ t♦ t ♠srt♦ ② r♥♥ s 32.8 ns ♥ t r♦♥ trt ② s 21.8 nsrs t ② ♦rrs♣♦♥♥ t♦ t ♠sr t♦ ② r♥♥ s22.17 ns r ♥ s♣rt♥ t r② ♣rt r♦♠ t t ♣rtr♣rs♥t t r♦♥ trt ②

st♦r♠s t♦ r♣rs♥t t ♥♠r ♦ Ps s ♣r t ♣♦st♦♥♦r t ♦♥rt♦♥ ♥♦♥ ♦♥② t ♦ ♦♥② t ♦r t ♦ ♥ ♦r ♥st♥ s♠t♦♥ ♥♦♥ ♦♥② Ps 15 ♦ st♠t ♣♦st♦♥ ♥ r③ t 5

s ♦ ♣♦st♦♥♥ rr♦r ♦♠♣rs♦♥ t♥ ♥ P ♦rt♠ss♥ ❯❲ ♦ P ♥ ❩ P s♦♥ ♦ ♦♥ Ps

rr♦r ♦ ♣♦st♦♥♥ s♥ P ♦r s ♥t♦♥ ♦ t ♠sr♠♥t♣♦♥ts ♦r ❯❲ ♦ P ♥ ❩ P s♦♥ ♦♦ ♥ Ps

t♦♥ ♦ ②♣♦tss tst♥ ♦♥ ♠sr t rr♦r ♦ ♣♦st♦♥♥ P s ②♣♦tss tst♥

r♣rés♥tt♦♥s ♠ê♠ r②♦♥ ♥s ①s ér♥ts s rrr ♣♦st♦♥♥♠♥t ♦♠♣r♥t P à s ♦rt♠s

t ❲ ♥÷ à ♣♦st♦♥♥r ♥ ♣♦st♦♥ B rç♦t ① ♦srs st♥s

r1 t r2 s ♥rs ① ♣♦st♦♥s R1 t R2 rrr ♦♠♠s sr sst♥s st ♠♦ésé ♣r ① ♥♥① ♦♥♥s ♥trés sr R1

t R2 ♥trst♦♥ s ♥♥① é♥èr s ré♦♥s rsés C1

t C2 rs r②♥trs C1 t C2 ♣s ♥÷ à ♣♦st♦♥♥rrç♦t s ♥♦r♠t♦♥s ♦ ♣ss♥ s ♥÷s ① ♣♦st♦♥sHk ♥ ♥♦t dk,i st♥ ♥tr Hk t Ci

rrr ♣♦st♦♥ ♥÷s ♥ts ♥ ♦♥t♦♥ ♥♦♠r ♥÷s ♥ts

st ♦ rs

strt♦♥ s r♣ ♣ès Gr r♣ ♠♥s Gw r♣ Gn t s rés① ♣rs♦♥♥s ss♦és à q ♥÷ ♦♠♣rs♦♥ ♥ ♦t♥ ♣r ♠sr t s♠é ♦t

rtr ré 21.8 ns t r ♠sré st 22.17 ns ♥ r♦ r♣rés♥t rtr ré

s rrr ♣♦st♦♥♥♠♥t ♦♠♣r♥t P

st ♦ s

st ♦ s

Pr♦♣♦s ♦♠tr t♦ ♦rt♠ sr♣t♦♥ Pr♠trs stt♥s s t♦ Pr♠trs stt♥s Pr♠trs stt♥s

❩ ♣t ♦ss ♠♦s ①trt ♦r s♥r♦ ❯❲ ♦ r♥♥ ♠♦ ①trt ♦r s♥r♦ s ♥ r♥ t♥ ♠srs ♥ s♠t♦♥s ♦r t

♥ r♥s ♦srs ttst ♣♦st♦♥♥ rr♦r ♦r s♥r♦ ttst ♣♦st♦♥♥ rr♦r P s P

sr♣t♦♥ s ét♣s ♥éssrs à st♠t♦♥ ♣♦st♦♥ Pr♠ètrs s♠t♦♥ Pr♠ètrs s♠t♦♥ Pr♠ètrs ♠♦è ♣t♦ss ①trt s♥r♦ Pr♠ètrs ♠♦è rrr st♥ ❯❲ ①trt s♥r♦ rrr qrtq ♠♦②♥♥ ♣♦st♦♥♥♠♥t P t P

st ♦ r♦♥②♠s

♦② r t♦rs ♠t ♥st② ♥t♦♥ ♥♥ ♠♣s s♣♦♥s r♠r♦ ♦r ♦♥ srt ♥t ♠t♦r ♥t r♥ ♠ ♦♠♥P ♦ t♦♥ ♦ Prs♦♥ ♦♠tr ♣ts❯❲ ♠♣s s♣♦♥s ❯tr ❲ ♥P ♦t♦♥ ♣♥♥t Pr♠tr ♥ ♦ t t ♠♥s♦♥ rr② ①♠♠ ♦♦♦ t♦ ♦ ♦♠♥ts❲ t ❲ ♥ ♦♦r ♦♥ ♥ ♦ tP P♦r ♥st② ♥t♦♥ ♦ ss ♥♦♦②P ♦st ♦♠tr P♦st♦♥♥ ♦rt♠ ♦♦t ♥ qr rr♦r ♥ tr♥t ②r♥♦ ♠ r♥ ♦ rr♦ ♠ ♦ rr❯ ❯♥♦r♠ ♦r② ♦ rt♦♥❯❲ ❯tr ❲ ♥❲ ❲t st qr

st ♦ t Pt♦♥s ss♦t t ts ❲♦r

st ♦ t Pt♦♥s ss♦tt ts ❲♦r

❬❯❪ ♠♦t r ♥ ❯♥ t♦♥ ♦ ♦♠tr ♣♦st♦♥♥ ♦rt♠ ♦r ②r rss ♥t♦rs ♦tr ♦♠♠♥t♦♥s♥ ♦♠♣tr t♦rs ♦t t ♥tr♥t♦♥ ♦♥r♥ ♦♥♣s s♣t

❬❯❪ ♠♦t r ♥ ❯♥ P②②rs ♥ ♣♥ ♦r ②♥♠♠t♦r ♦r ♥♦♦r Pr♦♣t♦♥ ♥ ♦③t♦♥ ♥tr♥t♦♥ ♦♥r♥♦♥ ♦♠♠♥t♦♥s ♠②

❬P❯❪ ♠♦t Prs♥ r ♥ ❯♥ ②r P♦st♦♥♥t♦ s ♦♥ ②♣♦tss st♥ ❲rss ♦♠♠♥t♦♥s ttrs PP

❬+❪ ♥♦t ♥s ♦♥ s r♥r ♥r r② r♥r ❯♥ ♦s ♠♦t ♦r♥ s ♦♦ ♦♠♥③ ♦sr ♦♠ r ♦r♥ ♠♥ t♥r ♥ rrt♥♦ ♦♦♣rt ♥ tr♦♥♦s ♥♦♦r ♦③t♦♥ ①♣r♠♥ts ♥ ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♦♠♠♥t♦♥s ❲♦rs♦♣ ♦♥ ♥s ♥ t♦r ♦③t♦♥ ♥ t♦♥ ❲♦rs♦♣ ♣st ♥r②♥

❬❯❪ r ♠♦t r♦♥ ♥ ❯♥ ♦rt ♣r♦r♠♥s ssss♠♥t ♦ ②r ♦③t♦♥ t♥qs ttst ♥ Pr♦ss♥ ❲♦rs♦♣ P ♣s ♥

❬❪ ♠♠② ss♥ s♥ ♦s ♠♦t ♥ t♥ s♥ rt♦♥♥ r♦ ♦ ♦r ♥♦♦r r♥ ♦ ♦ ❯srs ♥ sts t② ♣s

❬❯❪ ❯♥ ♠♦t ♥ r ①♣♦t♥ t r♣ sr♣t♦♥ ♦♥♦♦r ②♦t ♦r r② ♣rsst♥② ♠♦♥ ♥ ♠♦♥ ♥♥ ♣s ♠r

♦r♣②

♦r♣②

❬❪ ♥ ❨ ♠r s♦t♦♥ ♦ ♥t ♦♥r② ♣r♦♠s ♥♦♥♠①s qt♦♥s ♥ s♦tr♦♣ ♠ r♥s ♥t♥♥s ♥ Pr♦♣t♦♥♣s

❬❪ P♦rs ②sr ♥ ❲r♥r ❲s ❱rt♦♥ ♦ ②r ②r♥ ♦ ♦r ♥♦♦r ❯tr❲♥ ♥♥s ♥ r♦♣♥ ♦♥r♥ ♦♥ ♥t♥♥ ♥ Pr♦♣t♦♥ ♣s t

❬❪ ♦r rr♥t♦♥ ♦♠♣tt♦♥ ② ♦♠♥t t♦s ❲②

❬❪ ♦♥ ❲ ♥ ❨♥ r②tr♥ ♠t♦ ♦r ♠♦♥ ♥♦♦r ♣r♦♣t♦♥ ♥ ♣♥trt♦♥ ♥t♥♥s ♥ Pr♦♣t♦♥ r♥st♦♥s ♦♥ ♥

❬❪ ❩♥ ♥ ❨♥ ♥ ♥♦♥ ♥ ②r r♦rr ♠♦♠t ♦r♠t♦♥ ♦r tr♦♠♥t rt♦♥ ♣r♦♠s ♥ tr♦♠♥t ♦♠♣tt② ♥ t ♥tr♥t♦♥ ❩r ②♠♣♦s♠ ♦♥ tr♦♠♥t ♦♠♣tt② P sP ②♠♣♦s♠ ♦♥ ♣s

❬❪ ♦②♦♠♥ ♥ P Pt ♥♦r♠ ♦♠tr t♦r② ♦ rt♦♥ ♦r ♥ ♥ ♣rt② ♦♥t♥ sr Pr♦♥s ♦ t ♦

❬❪ ❲s ♥ ♥r② rt♦♥ t♦rt ♠♦ ♦ ❯ ♣r♦♣t♦♥♥ r♥ ♥r♦♥♠♥ts ♥t♥♥s ♥ Pr♦♣t♦♥ r♥st♦♥s ♦♥

❬❪ P ♦ss ② ❨ tt ♥ ♦s♥ r② ♥♥♠t♦ ♦r r♦♠♦ ♣r♦♣t♦♥ ♥ r♥ r ♥ ♥t♥♥s ♥ Pr♦♣t♦♥ ♦t② ♥tr♥t♦♥ ②♠♣♦s♠ P st ♣s ♦

❬❪ ❱③ ♦♦♣ ❯♥ ♠♦è rt♦♥ ♥ ♥s ♣r♠r ♣s♦ï rs♥ P tss ❯♥rsté P♦trs ♥r②

❬❪ ♥r t ♠♦ès ♣r♦♣t♦♥ sé sr té♦r rt♦♥ P tss ♥s ♥♥s ♥r②

❬❪ ♦♥♦ ♠ r r♥♦ ❲s ❱ r ♦rt♥ ❱♥③ ❲ ♦♠s ♥ ♥ ♦♦r ♦ ♣r♦♣t♦♥ ♠sr♠♥ts ♥ ♣rt♦♥ s♥ tr♠♥s♦♥ r② tr♥ ♥ r♥ ♥r♦♥

♦r♣②

♠♥ts t ③ ♥ ③ ❱r ♥♦♦② r♥st♦♥s ♦♥

❬❪ r ❲ö ♥r ♦♣♣ ♥ rr ♥st♦rr st ♥ ♥♥ r②♦♣t ♣r♦♣t♦♥ ♠♦ ♦r ♥♦♦r ♥ r♥ s♥r♦s s ♦♥ ♥ ♥t♥t♣r♣r♦ss♥ ♦ t ts ♥ Pr♦♥s ♦ t t ♥tr♥t♦♥②♠♣♦s♠ ♦♥ Prs♦♥ ♥♦♦r ♥ ♦ ♦ ♦♠♠♥t♦♥s Ptsr

❬❪ ❨ ♥ ♣♣♣♦rt ts♣ ♣r♦♣t♦♥ ♣rt♦♥ ♦r rss♥♥ ♣rs♦♥ ♦♠♠♥t♦♥ s②st♠ s♥ ❱r ♥♦♦② r♥st♦♥s ♦♥ ♦

❬❪ ♥ ❨♥ ♦♥ ❲ ♥ ♥② ♦ r②tr♥ ♠t♦ ♦r♠♦♥ ♥♦♦r ♣r♦♣t♦♥ ♥ ♣♥trt♦♥ ♥t♥♥s ♥ Pr♦♣t♦♥ r♥st♦♥s ♦♥

❬❪ ♠♦ss♦ ♦rr ♥♦r♠t♦♥ rt r♦♣♥ ♦♠♠ss♦♥ ❳ ♦♠♠♥t♦♥s ♥ ♦ sr t♦♥ t ♦ ♦♦rs tr ♥rt♦♥ ②st♠s ♥ ♣♦rt ❯ srs r♦♣♥♦♠♠ss♦♥

❬❪ ❱♦♥♦ ♦ ♣r♦♣t♦♥ ♠♦ s♦tr tt♣sr♦♠

♦♥♦s♦trsts♣① ss

❬❪ ❲♥Pr♦♣ ♦tr t ❲ Pr♦♣t♦♥ ♥ ♦ t♦r P♥♥♥ tt♣♦♠♠♥t♦♥s♦♠Pr♦ts ss

❬❪ tt♣r♠♦♠♦♠rss♥st

❬❪ ♥t ♦t♦♥s r tt♣s♥ts♦t♦♥s♦♠ ss

❬❪ rt③♥ ♥ P♥ ♠♦ ♦r s♣t ♦r ♦ ♠t♣t ♦♠♣♦♥♥ts ♣rt♥♥t t♦ ♥♦♦r ♦♦t♦♥ s♥ r② ♦♣ts ♥tr♥t♦♥♦r♥ ♦ ❲rss ♥♦r♠t♦♥ t♦rs ♣s

❬❪ r ♥ P♥ Pr♦r♠♥ t♦♥ ♦ ♥♦♦r ♦♦t♦♥s②st♠s s♥ PP♠ rr ♥ r② tr♥ s♦tr ♥ ❲rss ♦t♦rs ♥tr♥t♦♥ ❲♦rs♦♣ ♦♥ ♣s ② ♥

❬❪ r ♥♦♠♦ r♦s♦♣ Pstr♥ ♦ rtrsts ♦♣♠♥t ♦♥ ♠ Pr♦ss♥ t ♦t♦♥ ♥ ♠t♦♥ ♦ P tss ♦♦❯♥rst② ♣♥ ♥

❬❪ ③ ❩ ♥ ♥♥s ♦②s ♦s ❱ P♦♦r ♥❩ ♥♦ ♦③t♦♥ tr♥ r♦s ♦♦ t ♣♦st♦♥♥ s♣ts♦r tr s♥s♦r ♥t♦rs ♥ Pr♦ss♥ ③♥

❬❪ P ss♥r ♥ s ❲trs t♣tssst s♥♥♦r ♥♦♦r ♦③t♦♥ ♥ ♥ ♦ ♥r♦♥♠♥t ♥ ②st♠s ♥s ♥ ♠ Pr♦ss♥❲P t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣s

❬❪ P ss♥r ♥♠♥ ♥ r♥s♦ ♥ r t♥r r♦ sts ❩♠♥ ♥ ❲trs ♥ t s ♦ r② tr♥ ♦r ♣r♦r♠♥

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♦r♣②

♣rt♦♥ ♦ ♥♦♦r ♦③t♦♥ s②st♠s ♥ ❲♦rs♦♣ ♦♥♥s ♥ t♦r ♦③t♦♥ ♥ t♦♥ ♣st ♥r②

❬❪ rst♦♣ Prr ♦♠ ♥ ❨♥♥s P♦sst ♥ ♥ ♦♦♣ ❱③ ♥ ♥r r ❯tr❲♥ ♥♦♦r ♥♥ ♦♥ ❯s♥ ②r♥ ♦tr ♦r tr♦t❲ ♠♥ r♥tr♥t♦♥ ♦r♥ ♦ ♥t♥♥s ♥ Pr♦♣t♦♥ rt ♣s ♥

❬❪ ♦♦♥ ♥t ♥ rr♦♦ ♥t ♥s ♥ t ♥r♠♥tt♦r② ♦ rt♦♥ ♦r ♦♠♣① s♦r ♣♦♥t ♠♥t♦♥ ♥ ♥r ss♠②♥ ♥t ②♠♣♦s♠ ❳❳❳t ❯ ♣s

❬❪ st♥r ♦r ♥♦r♠t♦♥ t♥♦♦② t♦♠♠♥t♦♥s ♥ ♥♦r♠t♦♥①♥ t♥ s②st♠s ♦ ♥ ♠tr♦♣♦t♥ r ♥t♦rs s♣ rqr♠♥t ♣rt ❲rss ♠♠ ss ♦♥tr♦ ♠ ♥ ♣②s ②r♣② s♣t♦♥s ♦r ♦rt rss ♣rs♦♥ r ♥t♦rs ♣♥s t♦r

❬❪ r♣st♦ P♦ r♦♣♦♦s ♥ sts♥s ♥ ♦r ♦ t st♥r ♦♠♠♥t♦♥s ③♥

❬❪ ♣♦♦♥ ♥ t sts ❲♦r ♦r st♥ sr♠♥t ♦♥ ♥ ♣ tt♣s♣♦♦♥♦♠t

♦rr♦rst♥♦♥♦♥r♥ ss

❬❪ ❲ t♥♥ ♦ts t ♥ tt♣♦♠ ss

❬❪ st♥r ♦r ♦ ♥ ♠tr♦♣♦t♥ r ♥t♦rs ♣rt ❲rss♦② r ♥t♦rs t rr②

❬❪ ♦♥♥ s ♥ss♥ ♥♥ ❨♥ ♥ rst♥ rs ♣♣r♦ t♦ ♦ ♦♠♣①t② ❯❲ ♥♦♦r ♥ st♠t♦♥ ♥P

❬❪ ♦s♠t ❲rss ♦♠♠♥t♦♥s ♠r ❯♥rst② Prss st

❬❪ ♣♣♣♦rt ❲rss ♦♠♠♥t♦♥s Pr♥♣s ♥ ♣rt Pr♥t ♥r②

❬❪ ♦♠②♦♥ s♠ ♠♣s rs♣♦♥s ♠♦♥ ♦ ♥♦♦r r♦ ♣r♦♣t♦♥ ♥♥s ♦r♥ ♦♥ t rs ♥ ♦♠♠♥t♦♥s

❬❪ ♦rstr ts ♦ ♠t♣t ♥trr♥ ♦♥ t ♣r♦r♠♥ ♦ s②st♠s ♥ ♥ ♥♦♦r rss ♥♥ ♥ ❱r ♥♦♦② ♦♥r♥ ❱ ♣r♥ ❱ r ♦♠ ♣s ♦

❬❪ ♥ ❱♥③ sttst ♠♦ ♦r ♥♦♦r ♠t♣t ♣r♦♣t♦♥ ♦♠♠♥ ♣t♠r

❬❪ ♦♣♦s r♥s tr♦♠♥t ❲s ♥ ♥t♥♥s ♣tr trs❯♥rst②

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♦r♣②

❬❪ ♣♥trt♣ tt♣♦♣♥strt♠♣♦r ss

❬❪ st r♣ ♦r② tr♦♥ rr② ♦ ♠t♠ts ♣r♥r❱r

❬❪ t Pt♦♥ ♥ ♦rt♠ ♦r ♥♥ ♥♠♥t st ♦ ②s ♦ r♣♦♠♠♥ ♣t♠r

❬❪ ♥ ♦♥ ♦r♥ ♥ r♥s♦♥ ♥ ♥ ① ♥♦ ♥ ②♥♠ rt♦♥ ♥♦♦r ♥♥ ♠♦s ♦♥ r♦ ♣r♦ss ❲rss ♦♠♠♥t♦♥s r♥st♦♥s ♦♥

❬❪ Prs♥ ♥ r② ♦ ♥♥ ♦♥ ❯s♥ t♦st Pr♦♣t♦♥ r♣s ♥ ♦♠♠♥t♦♥s ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣s ♥

❬❪ s ♦♦ t♠♥s♦♥ tr① r ♥ t♠♥s♦♥ tr①s Prts ♥ ♣s Prss ②

❬❪ ♠P② tt♣♥♠♣②♦r ss

❬❪ ♦sr rt r♥r ❯♥ ♥ r♠♥ ♦♦♦♠ tr♠♥sts♠t♦♥ ♦ ❯❲ tr♥s♠ss♦♥ ♥♥ P②sq ♣t

❬❪ r r♦♥ ♥ ❯♥ ❱t♦r s♣r r♠♦♥s ♥t♥♥sr♣t♦♥ ♦r r r② tr♥ s♠t♦r ♥ tr♦♠♥ts ♥ ♥ ♣♣t♦♥s ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣s

❬❪ ❨ ♦♠ r té♣♥ r♦♥ r♥r ❯♥ ♥ ♥rt♥ ♥ t♣♥ Pr♦r♠♥ t♦♥ ♦ trs♦s t♦ st♠t♦♥t♥qs s♥ ❯❲ ♥♦♦r ♠sr♠♥ts ♥ ❲

❬❪ ♥♥ ♥ ♦t② ♦ ♦r ♥ ♥s rts ♦♠t♥♦♦② ♦r♥

❬❪ ♦rr t♦r ❲rss ① Prs♦♥③ ♦♠♠♥t♦♥s r♦♣♥ ♦♦♣rt♦♥ ♥ ♦ ♦ sr ♦♥ ❲② ♦♥s t♦♥ ♥ ♣♦rt

❬❪ ♦tt ♥ ♦r ♠t♥♦♦r ♠♦ ♦r ♥♦♦r r♦ ♣r♦♣t♦♥♦♠ ♦ s♥ r ②

❬❪ ♠ ssrt ♦ést♦♥ ♠tréq♥s ♥ ♣r♦♣t♦♥P tss ❯♥rst② ♦ P♦trs ♠r

❬❪ ♠t♥s P ♦③t♦♥ ♦rt♠s ♥ rss s♥s♦r ♥t♦rs rr♥t♣♣r♦s ♥ tr ♥s t♦r Pr♦t♦♦s ♥ ♦rt♠s

❬❪ r ♦♠♠♥t♦♥s ♦♠♠ss♦♥ ♥s ♦r st♥ ♥ ❱r②♥ tr② ♦ ❲rss ♦t♦♥ ②st♠s ♣♦rts ♥ rrr t♣r

❬❪ ❲ Pr♦t r ♦t♦♥ s ♦♣t♠st♦♥ ♦rP❨ ♦rt♠s♣r♦t♦♦s ②

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♦r♣②

❬❪ Ptr ♦s s ♣②r♦s ②♣r♦♥ts r r♦ ♥♦♣ ♦ss ♥ ②r ♦rr ♦t♥ t ♥♦s ♦♦♣rt ♦③t♦♥ ♥rss s♥s♦r ♥t♦rs ♥ Pr♦ss♥ ③♥

❬❪ r ♦ ❱ ♥ P r♥♦s ♦ tr♠♥ ♣♦st♦♥♥ ♣♦r ② ♣r♦ ♥r♣r♥t♥ ♣r♦ t♦♥ s♠t♦♥s Pr♦♥s ♦ t ❱r ♥♦♦② ♦♥r♥ ❱ ♣s ♣t♠r

❬❪ t♥♥ t♥♠ ♥ ♦rstr♦♠ ts ♦rrt♦♥ t♦♦r ♦t♦♥ Pr♦♥s ♦ t ❱r ♥♦♦② ♦♥r♥❱♣r♥ ②

❬❪ ❲ss ♥ t r② ♦ r ♦t♦♥ ②st♠ s ♦♥ sr♠♥ts tr♥st♦♥s ♦♥ r t♥♦♦②

❬❪ ♦r♦r ❨ ♦st♥♥ ♥ ❱ ss s ♥♦♦r ♦③t♦♥ t ♠t♣♦②♠♥t ♦ ♥ ♦ ♦♠♠♥t♦♥s ♦♥r♥ ♣s

❬❪ ③ sr② ♦♥ rss ♣♦st♦♥ st♠t♦♥ ❲rss Prs♦♥ ♦♠♠♥t♦♥s ♣r♥r tr♥s rr②

❬❪ ♥♥r s ♥r ♥s ♥ ❯♥ r ♥ ♦tt♥ ♦st ♥♠♥t Pt♦r♠ ♦r t♥s♦r♦t♦♥ t ♥tr♣rtt♦♥ Pr♦♥s ♦ ♠♦ s♠♠t ♥

❬❪ ♦♠ r ❨ t♣♥ r♦♥ ♥ r♥r ❯♥ ♦♠♣rs♦♥ ♦ ②r ♦③t♦♥ ♠s s♥ ♥ ❲rss♦♥r♥ st♥ ❲rss ♥♦♦s r♦♣♥ ❲rss tr♦♣♥ ♣s ♣r

❬❪ ♦rt② ♦ ♥ Pstr ♦♥① ♣♦st♦♥ st♠t♦♥ ♥ rsss♥s♦r ♥t♦rs Pr♦♥s ♦

❬❪ rs ② ♥ t ♦t♦♥♥ ♥ strt ♦❲rss ♥s♦r t♦rs Pr♦♥s ♦ ♥tr♥t♦♥ ♦♥r♥ ♦♥♦sts ♣ ♥ ♥ Pr♦ss♥

❬❪ ♦♥ ♥♥♦♥ ♥ s rs strt ♦③t♦♥ ♥ rss s♥s♦r♥t♦rs q♥ttt ♦♠♣rs♦♥ ♦♠♣t t ♦♠r

❬❪ rst♦s P♦ts ♦s♥ r ①t ♥rs r ♦♥❨♦♥ ♠♠♦ ♠ ❯s ♥ ♠ ③♦ ♦♦♣rt t♦rs♦r t tr ❲rss ❲♦r ♦♠♠♥t♦♥s ③♥ ♣t♠r

❬❪ t♥♦ r s♣♣ r s♣♣ st♥♦ ♥ rr t♦♣rss ♠ss♣ss♥ ♦♠♣①t② ♥ r② ♦♠♣rs♦♥ ♦r strt♦③t♦♥ ♥ ❲P ♣s

❬❪ r♥ ss♥ r♥♥ r② ♥ ♥s♥r ♦r t♦r r♣s♥ t s♠♣r♦t ♦rt♠ ❨

♦r♣②

❬❪ ❯ r♥r ❲②♠rs ♥ ❩ ❲♥ ♦♦♣rt ♥♦rss ♦③t♦♥ ♦rr ②♥♠ ♥t♦rs ♥ ❯tr❲♥ ❯❲ ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♦♠ ♣s

❬❪ ❨ r♦♠ ♥ ❳ st♠t♦♥ ♥ r♥ Pr♥♣s ♥qs♥ ♦tr ❨s Ps♥ ♥♦♥ t♦♥ rr②

❬❪ r ♦♥trt♦♥ ♦♥ ②r ♦③t♦♥ ♥qs ♦r tr♦♥♦s❲rss t♦rs P tss ❯♥rst② ♦ ♥♥s ♠r

❬❪ r r♦♥ ♥ ❯♥ r♦♠♥ ♥rts ♥ ♦s ♦t♦♥ st♠t♦♥ ❯s♥ ♦t st qr Pr♦♥s ♦ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♥s rts ♥ ②st♠s ♣s ♦♠r

❬❪ ♦♦♥ ①♣tt♦♥♠①♠③t♦♥ ♦rt♠ ♥ Pr♦ss♥ ③♥

❬❪ ♦ t♠t ttsts ♥ t♦♥ ♣r♥r t♦r

❬❪ ♠♦♥ ♦♦r ♥tr ♥②ss

❬❪ ♦♥ ♥s♥ ♥tr rt♠t ♥ ♠tr① ♦♠♣tt♦♥s ♣rt

❬❪ rt♥ r♦ ❱ r♥♦ ♥ ♥③r ♥ t s ♦ ♥trs♥ s♥t ♦♠♣t♥ ❲t s t st tr♥st♦♥ r♦♠ ♥r t♦ qrt♣♣r♦①♠t♦♥ ♥ ♦♥rr s♥ ♥ r③② ❲s♥s t♦rsP ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr ♥ ♣s ♣r♥r

❬❪ ♦♥③ r♠♥ts ❱r③♥♥ ♥ rs ttst♥ ♦t ♦♠♣t♥ ♣♣r♦s ♥ ♥sr♥ Pr♦♠s ♦♠♣tt♦♥ t♠ts ♥ ♥②ss ♦ ♥ Psrs ♥♦r♣♦rt

❬❪ ❲ ♠♥ ss♥② P ♠♥ t♥ r rr♥t②t ♥ s♦r s♦t♦♥ t♦ t s♠t♥♦s ♦③t♦♥ ♥ ♠♣♥ s♠ ♣r♦♠ r♥st♦♥s ♦♥ ♦♦ts ♥ t♦♠t♦♥

❬❪ s ♠♥ ♥ ❨ ♥ t ♠♠rs♣ stt ♥ ♣r♠trst♠t♦♥ ♦r s②st♠s sr ② ♥♦♥♥r r♥t qt♦♥s t♦♠t t♦r

❬❪ ♠♦ ♥♥ ♥ P♣♣ ♦♥♥t ♦♥str♥ts ♣r♦♣t♦♥ t♥qs♦♥ ♥trs ♦r r♥t ♦③t♦♥ s♥ r♥♥t t t♦♠t

❬❪ r rs ♥②s ♣r ♥trs ♣♦r ♦st♦♥ t rt♦r♣ s♠t♥és ♣♣t♦♥ à r♦♦tq s♦s♠r♥ s ❯♥rsté rt♥♦♥t rst t♦r

❬❪ ❱ r ♥ P ♦♥♥t ♥trt② ♦t♦♥ ③♦♥ rtr③t♦♥ s♥ ♥tr ♥②ss ♥♥ ❯♥t tts

♦r♣②

❬❪ ❱ r ♥ P ♦♥♥t ♦ ♣♦st♦♥♥ ♥ r♥ rs t ♠♣s♥ ♥t♥t ❱s ②♠♣♦s♠ ❱ ♣s ♥

❬❪ ♥ ②♥t ♥ rt ♦st t♦ ♣ss ♦t♦♥ s♥ ♥tr♥②ss ♥ ♦♥trt♦r ♣r♦r♠♠♥ ♥ r ♦r① r♥

❬❪ r ♦r ♠ ♥♦ss ♥ ér r r♥t♦① ♦③t♦♥ ♥ s ② ♥tr ♥②ss ♥ ♦♥str♥ts Pr♦♣t♦♥ ♥qs ♥ ♦ ♦♠♠♥t♦♥s ♦♥r♥

❬❪ ♦r ♥♦ss ♥ r ♦st ♦③t♦♥♦rt♠ ♦r ♦ ♥s♦rs ❯s♥ ♥t♦♥s ❱r ♥♦♦② r♥st♦♥s ♦♥ ♠②

❬❪ ♥♥♦st r ♥ r r♥t r♦st strt st♠t♦♥♥ ♥t♦r ♦ s♥s♦rs ♥ P ♣s

❬❪ ♥ r rt ♥ ❲tr ♣♣ ♥tr ♥②ss ♣r♥r♣t♠r

❬❪ s rt ♥qs ♥trs ♣♦r rés♦t♦♥ s②stè♠séqt♦♥s P tss ♦♣ ♥t♣♦s ès ♦t♦rt ♥♦r♠tq

❬❪ ❱ r ♥ P ♦♥♥t ♣♦st♦♥♥ ♦♠♥ ♦♠♣tt♦♥ ♦r r♥♥t♦♥ ♥t♥t r♥s♣♦rtt♦♥ ②st♠s ③♥

❬❪ r r♦♥ ♥ ❯♥ ②r t s♦♥ t♥qs ♦r♦③t♦♥ ♥ ❯❲ ♥t♦rs ♥ P♦st♦♥♥ t♦♥ ♥ ♦♠♠♥t♦♥ ❲P t ❲♦rs♦♣ ♦♥ ♣s ♠r

❬❪ ♥r ❲ ♦♦r ♥ ♥t♦t♦r② tt♦r ♦♥ trs

❬❪ r r♦♥ ♥ r♥r ❯♥ ♥♥♥ ♣♦st♦♥♥ r②tr♦ rt ♣♦st♦♥ st♠t♦rs s ♦♥ ②r t s♦♥ ♥ Pr♦❱ ♣r♥ ♣s r♦♥ ♣♥

❬❪ ❲ Pr♦t r sr♠♥ts ♦ ♦t♦♥♣♥♥t ♥♥ trs t♦r

❬❪ ② ♥♠♥ts ♦ ttst ♥ Pr♦ss♥ tt♦♥ t♦r②Pr♥t P

❬❪ P♣♦♥st♥t♥♦ ♥ ♦ r♠r♦ ♦♥s ♦r ②r ♦③t♦♥♠t♦s ♥ ♦ ♥ ♦ ♥r♦♥♠♥ts ♣s s♣t

❬❪ r r♦♥ ♥ ❯♥ ①♠♠ ♦♦ ♦ sst♠t♦r ♦r ♦③t♦♥ ♥ tr♦♥♦s t♦rs ♥tr♥t♦♥ ♦r♥♦♥ ♦♠♠♥t♦♥ t♦r ♥ ②st♠ ♥s ♥r②

❬❪ ❯ ❯❲ ♦♠ ♥ Pr♦ss♥ ♥qs ♦r❲rss ♦♦♣rt ♦③t♦♥ ♥ r♥ s éé♦♠ rt♥ ❯♥rsté rt♥ ♥t ♦♠r

♦r♣②

❬❪ qs P♦t ♦s♣ t♦ ❩♥r ♥ rr ♦♥r ♣ss ♦♥ ②rs ♦ ♦♥t r♦ stt♦trt ♥ ♣rs♣ts ❯P ❲rss ♦♠♠ ♥ t♦r♥

❬❪ ♥rés r rr r♦♥ ♦♠s rt ♦♥ tt② ♥ ② r ♥ t ♦st♥ss ♦ ♠♣ ♥♦♦r ♠t♦♥ ♦s ♦ s♥s♦r rss ♥t♦rs

❬❪ ♠② ❲ s ♦t② ♦♥ ♦r ❱r t♦rs ♣♠♥

❬❪ t♥ t♠♥ ♦♥ r♦♥t ♥ P♣♣ ♥ ②♥ ♥ ♦ ♦t♦rs r♦♥♥ ♦t♦♥ ♦t② ♦ ♥ ♣♦rt ♣t♠r

❬❪ ♦ ♥♠♥ts ♦ rss ♥t♦rs ♣tr ♣s sr♥ ♠str♠ tr♥s

❬❪ ♥♦♥ ♥s ♥ ♦♥ ♦♥ ②♥♥ ♦♥ ♦♦♥ ♠ ♥♦♥ ♦♥ ♥ t ② ♥tr ♦ ♠♥ ♠♦t② r♥st ♥

❬❪ ♠♥ P♦r s ♣rt♦ strt♦♥s ♥ ③♣s ♦♥t♠♣♦rr②P②ss ♠r

❬❪ ❨♥ ♥ï ♥ t ❯s♥ r♠♥ ♥ s♠♦ t♦ st②♣r♦t♦♦ ♦r ♠♦ ♠t♦♣ ♥t♦rs ♥ ♣s

❬❪ ♥ ä♥r r t♣♥♦ ♥ ♦tr♠ r♣s♠♦t② ♠♦ ♦r ♠♦ ♦ ♥t♦r s♠t♦♥ ♥ ♠t♦♥ ②♠♣♦s♠ Pr♦♥s t ♥♥ ♣s

❬❪ ♥ r②♥♥ ♦♣♥ ♥ ♠ ♠② P r♠♦r t♦ s②st♠t② ♥②③ t ♠♣t ♦ ♦t② ♦♥ Pr♦r♠♥ ♦ ♦♥ ♣r♦t♦♦s ♦r ♦ ♦rs ♥ ♥t②♦♥ ♥♥♦♥t ♦♥r♥ ♦ t ♦♠♣tr ♥ ♦♠♠♥t♦♥s ♦ts♦♠ ♣s

❬❪ ♠t r♦s ③t ♥♦②r ♥ ♠r♦t ♥ sr ♦rs rst ♠♦t② ♠♦s ♦r ♠♦ ♦ ♥t♦rs ♥ Pr♦♥s♦ t t ♥♥ ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♦ ♦♠♣t♥ ♥ ♥t♦r♥♦♦♠ ♣s ❨♦r ❨ ❯

❬❪ ❲②♠rs ♥ ♥ ❩ ❲♥ ♦♦♣rt ♦③t♦♥ ♥ ❲rsst♦rs Pr♦♥s ♦ t

❬❪ rt ♦♥③③ sr ♦ ♥ rts③♦ rs ❯♥rst♥♥♥ ♠♥ ♠♦t② ♣ttr♥s tr

❬❪ r♥ ♠s ♥ t♣♥ t Pr srt♥t s♠t♦♥ rst♦rs Prs♦♥ Pr♥t ❯♣♣r r

❬❪ ♥ r③③ ♦ r♠♥♥ rs ♥ r r ♥t♦♣♠♥t ♥ ♣♣t♦♥s ♦ ❯ ♠t♦♥ ♦ ❯r♥ t② ♥tr♥t♦♥ ♦r♥ ♥ ♥s ♥ ②st♠s ♥ sr♠♥ts ♠r

♦r♣②

❬❪ r♥♦ é♥ ♦s♦s ♦♥r♦②t♦♥ ♥ ❨ r②s♥t♦ ♥t♦r s♠t♦r s ♣t♦r♠ ♦r r♥ ♦r ♦♣♠♥t

❬❪ ♦ rst ♥t♦r s♠t♦r tt♣♠♦r♥t ss

❬❪ ♠P② ♠t♦♥ P tt♣s♠♣②s♦r♦r♥t ss

❬❪ r ❲ ②♥♦s tr♥ ♦rs ♦r t♦♥♦♠♦s rtrs ♥ ♠♦♣rs ♦♥r♥ ♦♠ ♣s

❬❪ Prs♦♥ ♣ r♥st ♠t♦♥ tt♣s♦r♦r♥t♣r♦ts♣rtss

❬❪ ♠❲ Pstr♥ ♠t♦♥ ♦r r♥s♣♦rt t♦♥ ♥ rttr tt♣s♠♦♠ ss

❬❪ P❱ ❱s tt♣s♦♥tr♣tr♦♣♦♠♥s♣r♦ts

♣ts ss

❬❪ ♦♥ Pstr♥ ♦tr tt♣♦♥♦♠ ss

❬❪ t♦r ♠t♦r ♥s tt♣s♥s♥♠♥s ss

❬❪ st♦ r♥r♦ r ♦♥ts ♥ ♥ r♦ st ♣r♦t♦t②♣♥ ♦ ♥t♦r ♣r♦t♦♦s tr♦ ♥s s♠t♦♥ ♠♦ rs ♠t♦♥ ♦♥ Prt ♥ ♦r②

❬❪ ❲t ❲♦rs♥s s♠t♦r tt♣s♥t♦r♥rr ss

❬❪ ❱t♦r ♥ r② r r t♦♠t ♠r♦s♠t♦♥ ♦r ♠♦♥ rt♦♥ ♣str♥ ②s r♥s♣♦rtt♦♥ sr Prt t♦♦♦ r

❬❪ ♥ str rr② P ♠♠r♠♥s ♥ ♦r♥ ssr♥ ♠t♥t rt♦♠t s②st♠ ♦r ss♥ s♠t ♣str♥ tt② ♥ t♥ ♥♥♥ ♦♠s ❲♦rs t♦rs ♣s ♣r♥r

❬❪ ❨♥ ♥ï ♥ t ❯ ♠t♦♥ ♥ ♦r❯sr ♦t② ♦s ♥t♦♥ ♥ t♦♣ t♦rs ♥ rst sP Pr♦r♠♠♥ ♥s ♥ ♦♠♣rs ❲♦rs♦♣ PP ♣ ♥♥ ♥

❬❪ ♦s ♥s♥ ♥ ♥ r♥r ♠♣t ♦ rst ♥♦♦r ♠♦t②♠♦♥ ♥ t ♦♥t①t ♦ ♣r♦♣t♦♥ ♠♦♥ ♦♥ t sr ♥ ♥t♦r ①♣r♥ ♥ ♥t♥♥s ♥ Pr♦♣t♦♥ P t r♦♣♥ ♦♥r♥♦♥ ♣s

❬❪ ♠♦ r ❯②♠ ❯♠ r♦③♠ ❨♠ ❨s♠♦t♦ ♥ r♦ s♥♦ ❯r♥ ♣str♥ ♠♦t② ♦r ♠♦ rss♥t♦r s♠t♦♥ ♦ t♦rs

❬❪ ❲ str ♦rt ♥tr♦t♦♥ t♦ t rt ♦ Pr♦r♠♠♥ ♦♥

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♦r♣②

❬❪ r ♥tr♦t♦ ♥ ♥②s♥ ♥♥t♦r♠ ♥tr♦t♦ ♥ ♥②s♥ ♥♥t♦r♠♣ r♠♠ ♦sqt s♦♦s

❬❪ ♥ ♠♥s♦♥ ♦②

❬❪ r♦s♦♣ Pstr♥ ♠t♦♥ ♦ t♦ t ❵♥ rt♦♥♦Pstr♥ r♦ss♥ ♥ Pr♦♥s ♦ ♥rstrtr P♥♥♥ ♦♥r♥♦ ♣♥

❬❪ s Prs♥ r♦s Prs♥ ♥ r♥r r② ①♣♦t♥ t♦r♦♣♦♦② ♥♦r♠t♦♥ t♦ tt ♠ts ♥ ❱P ♦③t♦♥ ♥ t ♥tr♥t♦♥ ❲♦rs♦♣ ♦♥ ♦♠♣tt♦♥ ♥s ♥ t♥s♦r ♣t Pr♦ss♥ P ♣s ♥ ♥ Prt♦ ♦ ♠r

❬❪ s ♥ ❲②♠rs ♥s♦r♥ ♦r ②s♥ ♦♦♣rt P♦st♦♥♥ ♥♥s ❲rss t♦rs t rs ♥ ♦♠♠♥t♦♥s ♦r♥ ♦♥

❬❪ ❩rr ♥ ♥s ♦♠♣rs♦♥ ♦ ♥s st♦♥ rtr ♦r ♠♦ tr♠♥♣♦st♦♥♥ ♥ ♦♦♣rt tr♦♥♦s ♥t♦rs ♥ ♦tr ♦♠♠♥t♦♥s ♥ ♦♠♣tr t♦rs ♦t t ♥tr♥t♦♥ ♦♥r♥♦♥

❬❪ ❲ ❲rss ②r ♥♥ ♦ ♦ st♠t♦rs Ps tt♣♥srr ss

❬❪ P♦st ♣r♦ss s s tt♣ssr♦♠rss ss

❬❪ ♥tr♦♣rt② r♠♦r rqr♠♥ts ♥ ♠♥ s♣t♦♥s P ❲ r t♦r

❬❪ P③③♥ ♥ rtr ♦ ♣♦r ♦ t rt ♠♣s r♦ tr ♥ tr♥sr ♥ tr t♦r ♥ ♦ ♠♠t ♣s

❬❪ st ♥ t♦♥ ♦ t ♥trt s②st♠ ♥r r s♥r♦ ♦♥t♦♥s P ❲ r t♦r

❬❪ s ♥ Pr♠♦ ❱s♥t ♥♠♥ts ♦ ❲rss ♦♠♠♥t♦♥♠r ❯♥rst② Prss ♥

❬❪ r ♠♦t r♦♥ ♥ ❯♥ ♦rt ♣r♦r♠♥s ssss♠♥t ♦ ②r ♦③t♦♥ t♥qs ttst ♥ Pr♦ss♥ ❲♦rs♦♣P ♣s ♥

❬❪ P②trs tt♣♣②②rs♦r ss

❬❪ rt ♥st♥ ♦♥t♦♥ ♦ t ♥r t♦r② ♦ rtt② ♥♥♥ rP②s

❬❪ P②t♦♥ Pr♦r♠♠♥ ♥ tt♣♣②t♦♥♦r ss

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strt

s tss ♦ss ♦♥ t ♦♣♠♥t ♦ t♦♦s ♥ ♠t♦s t ♦r tr ♥❯❲ ♦③t♦♥ s②st♠s ♥ ♥♦♦r ♥r♦♥♠♥t tss ♦r s ♦♥t t♥ tr♦♣♥ P ♣r♦t ❲r ♦t t ♦♦♣rt ♦③t♦♥ ♥ r ♥t♦rs tr♦♠ ♠sr♠♥t ♠♣♥ ♦♥t r♥ t ♣r♦t r s t♦ t t ♣r♦♣♦s♦rt♠s s tss s ♥ ♦r ♣rts

rst ♣rt s ♦s ♦♥ t sr♣t♦♥ ♦ ♥ ♦r♥ r②tr♥ t♦♦ s ♦♥ r♣sr♣t♦♥ ♥ ♦rr t♦ ♦♠♣♥t t t rqr♠♥t ♦ ♠♦ s♠t♦♥ ♥♦♥♣t ♦ r②s s♥tr ♥♥ ♥r♠♥t ♦♠♣tt♦♥ ♥ t♦r③ ♦r♠s♠ ♦r♣r♦ss♥ r②s r sr ♥ ♠♣♠♥t s♦♥ ♣rt s ♦s ♦♥ t ♥♦♦r ♦③t♦♥ t♥qs r ♥♦ t♥q s ♦♥ ♥tr ♥②ss ♣♣r♦s s ♣rs♥t♥ ♦♠♣r t♦ tr♥t t♥qs ♥t♦s② s♥ ts ♣♣r♦ s♣ ♣r♦ss♥ s ♦♥ ♥ ②♣♦tss tst♥ ♠t♦ s♥ r ♣♦r ♦srt♦♥s t♦ rs♦♠ts ♣♣r♥ ♥ ♥r tr♠♥ ♦③t♦♥ ♣r♦♠s s sr tr ♣rtsrs r♥t s♣ts ♦ t ②♥♠ ♣t♦r♠ ♥ ♣rtr rst ♠♦t② ♠♦s ♦♥ str♥ ♦rs r♣ sr♣t♦♥ ♦ t ♥t♦r s♥ ♥ ♥ ♥tr ♥ts♦♠♠♥t♦♥ ♣r♦t♦♦ r t ♦rt st♦♥ ss ♠sr t ♦t♥ r♦♠ ♥tr♦♥♦s ♠sr♠♥t ♠♣♥ t♦ t ♦t t ♦♣ s♦tr ♣t♦r♠ ♥t ♣r♦♣♦s ♦③t♦♥ ♦rt♠s

és♠é

tt tès ♣♦rt sr é♦♣♣♠♥t ♦ts t ♠ét♦s ♣♦r ét s s②stè♠s ♦st♦♥ ❯tr r ♥ ♥ ♠ ♥térr tr tès été ♠♥é ♣♦r ♣rt♥s r ♣r♦t r♦♣é♥ P ❲ ♣♦rt♥t sr ♦st♦♥ ♦♦♣ért ♥ss rés① rs tès ts ♣♦r s ♣rt t♦♥ s ♦♥♥és ♦t♥s ♥s r ♣r♦t tès ♦♠♣♦rt r♥s ♣rts

❯♥ ♣r♠èr ♣rt ♣rés♥t ♥ ♦t r②tr♥ sé sr ♥ sr♣t♦♥ à s r♣s ♥ ♣♦♦r rssr s ♣r♦é♠tqs s♠t♦♥ ♠♦té ♦t ♥tr♦t ♦♥♣t ♥♦ s♥tr ♥s q♥ ♦r♠s♠ t♦rsé ♣r♠tt♥t éért♦♥ ♠♣ sr s r②♦♥s ♦t♥s ❯♥ s♦♥ ♣rt ♦♥r♥ s t♥qs ♦st♦♥ tsés ♥ ♥térr t ♣r♦♣♦s ♥ ♠ét♦ ♦r♥ sé sr s ♣♣r♦s♥s♠sts tt ♠ét♦ st éé t ♦♠♣ré à s t♥qs tr♥ts ♦♠♠ s ♠♦♥rs rrés ♣♦♥érés ♦ ♠①♠♠ rs♠♥ r♥t ♣rt s s♣étés ♠ét♦ ♣réé♥t ♥ ♠ét♦ sé sr ♥ tst ②♣♦tès st ért tt r♥èr♣r♦♣♦s ①♣♦tr s ♦♥♥és ♣ss♥ rç r♠♥t s♣♦♥ ♥ ♣rtq ♣♦r rs ♠ïtés ♠t♠♦s ♥s s s r♥ ♦srs ♣rés ❯♥ tr♦sè♠ ♣rt♣rés♥t s♣ts ♣t♦r♠ ②♥♠q ♦t ♦r ♥ s♠t♦♥ ♠♦térést sé sr s str♥ ♦rs ♣s sr♣t♦♥ s♦s ♦r♠ r♣ réss♥s t ♥♥ ♥ ♣r♦t♦♦ s♠♣é ♦♠♠♥t♦♥ ♥tr ♥ts qtrè♠ ♣rt①♣♦t s ♦♥♥és r♦ ♦t♥s ♦rs ♥ ♠♣♥ ♠sr ♣♦r r s ér♥tséts ♣t♦r♠ t s ♦rt♠s ♦st♦♥ ♣r♦♣♦sés

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