●❛❜r✐❡❧ ❍✐❞❡q✉✐ ❊r❜❛♥♦
❆♥á❧✐s❡ ❞❡ ❙ér✐❡s ❞❡ ❚❡♠♣♦ ❋✐♥❛♥❝❡✐r❛s
❯♠❛ ❆♣❧✐❝❛çã♦ ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s ❡♠
❋✐♥❛♥ç❛s ❊♠♣ír✐❝❛s
●❛❜r✐❡❧ ❍✐❞❡q✉✐ ❊r❜❛♥♦
❆♥á❧✐s❡ ❞❡ ❙ér✐❡s ❞❡ ❚❡♠♣♦ ❋✐♥❛♥❝❡✐r❛s
❯♠❛ ❆♣❧✐❝❛çã♦ ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s ❡♠
❋✐♥❛♥ç❛s ❊♠♣ír✐❝❛s
❆tr❛✈és ❞♦ ✉s♦ ❞❡ té❝♥✐❝❛s ♦r✐❣✐♥❛❧♠❡♥t❡ ❡♠✲ ♣r❡❣❛❞❛s ♣❡❧❛ ❋ís✐❝❛✱ ❜✉s❝❛✲s❡ ✐❞❡♥t✐✜❝❛r ❝❛r✲ ❛❝t❡ríst✐❝❛s ❞❡ ❝♦♠♣♦rt❛♠❡♥t❡ ❞❡t❡r♠✐♥ís✲ t✐❝♦ ❡♠ sér✐❡s ❞❡ t❡♠♣♦ ✜♥❛♥❝❡✐r❛s✳
❖r✐❡♥t❛❞♦r✿
Pr♦❢✳ ❉r✳ ❲✐❧❧✐❛♠ ❊✐❞ ❏r✳
❊s❝♦❧❛ ❞❡ ❆❞♠✐♥✐str❛çã♦ ❞❡ ❊♠♣r❡s❛s ❞❡ ❙ã♦ P❛✉❧♦ ❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ s♦❜ ♦ tít✉❧♦ ❆♥á❧✐s❡ ❞❡ ❙ér✐❡s ❞❡ ❚❡♠♣♦ ❋✐♥❛♥❝❡✐r❛s ✕ ❯♠❛ ❆♣❧✐❝❛çã♦ ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s ❡♠ ❋✐♥❛♥ç❛s ❊♠♣ír✐❝❛s✱ ❞❡❢❡♥❞✐❞❛ ♣♦r ●❛❜r✐❡❧ ❍✐❞❡q✉✐ ❊r❜❛♥♦ ❡ ❛♣r♦✈❛❞❛ ❡♠ ✶✺ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✵✹✱ ❡♠ ❙ã♦ P❛✉❧♦✱ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ ♣❡❧❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❝♦♥st✐t✉í❞❛ ♣❡❧♦s ❞♦✉t♦r❡s✿
Pr♦❢✳ ❉r✳ ❲✐❧❧✐❛♠ ❊✐❞ ❏r✳ ❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❆❜r❛ã♦ ▲❛r❡❞♦ ❙✐❝s✉ ❋✉♥❞❛çã♦ ●❡tú❧✐♦ ❱❛r❣❛s
❆❣r❛❞❡❝✐♠❡♥t♦s
❉❡❞✐❝♦ ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣❛r❛✿
✕ ♦ ♣r♦❢❡ss♦r ❞♦✉t♦r ❲✐❧❧✐❛♠ ❊✐❞ ❏r✳✱ ♣❡❧♦ s❡✉ ❛♣♦✐♦ ❡ ❡♥❝♦r❛❥❛♠❡♥t♦❀
✕ ♦ ♣r♦❢❡ss♦r ❞♦✉t♦r ❘✐❝❛r❞♦ ❘♦❝❤♠❛♥✱ ♣❡❧❛ s✉❣❡stã♦ ❞♦ t❡♠❛❀
❘❡s✉♠♦
❆❜str❛❝t
❙✉♠ár✐♦
✶ ❖❜❥❡t✐✈♦s ❡ ❏✉st✐✜❝❛t✐✈❛s ♣✳ ✶✵
✶✳✶ ❈♦♥❝❡✐t♦s ❋✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✹
✶✳✶✳✶ ❈❛♦s ❡ ❈♦♠♣♦rt❛♠❡♥t♦ ❈❛ót✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✹
✶✳✶✳✷ ❆tr❛t♦r❡s ❊str❛♥❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✹
✷ ❍✐♣ót❡s❡s ❡ ▼❡t♦❞♦❧♦❣✐❛ ♣✳ ✶✻
❍✐♣ót❡s❡ ✶✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✻
✷✳✶ ❉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✼
✸ ❘❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ♣✳ ✷✸
✸✳✶ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❞❡❢❛s❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✺
✸✳✶✳✶ ❊①♣❛♥sã♦ ❡ ❈♦♥tr❛çã♦ ❞♦ ❆tr❛t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✻
✸✳✶✳✷ ■♥❢♦r♠❛çã♦ ♠út✉❛ ♠é❞✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✻
✸✳✷ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶
✸✳✷✳✶ ❖ ♣r♦❜❧❡♠❛ ❞♦s ❢❛❧s♦s ✈✐③✐♥❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷
✹ ❈❧❛ss✐✜❝❛♥❞♦ ♦ ❙✐st❡♠❛ ❉✐♥â♠✐❝♦ ♣✳ ✸✺
✹✳✶ ❉✐♠❡♥sõ❡s ❋r❛❝t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✺
✹✳✶✳✶ ❈♦rr❡❧❛çõ❡s t❡♠♣♦r❛✐s ❡ ❣rá✜❝♦s ❞❡ s❡♣❛r❛çã♦ ❡s♣❛ç♦✲t❡♠♣♦r❛❧ ✳ ♣✳ ✸✾
✹✳✷ ❊①♣♦❡♥t❡s ❞❡ ▲②❛♣✉♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✵
✹✳✸ ❊♥tr♦♣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸
✺ ❆♥á❧✐s❡ ❞❛s ❙ér✐❡s ♣✳ ✹✻
✺✳✶ ■❜♦✈❡s♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✼
✺✳✷ ■❇r❳✲✺✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✵
✺✳✸ P❡tr♦❜rás P◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✺
✺✳✹ ❊♠❜r❛t❡❧ P❛rt✐❝✐♣❛çõ❡s P◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✾
✺✳✺ ❈â♠❜✐♦ ❘❡❛❧✲❉ó❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✷
✻ ❈♦♥❝❧✉sã♦ ♣✳ ✻✺
✼ ●❧♦ssár✐♦ ♣✳ ✻✼
❆tr❛t♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
❈❛♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
❉✐♠❡♥sã♦ ❞❡ ✐♠❡rsã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
❉✐♠❡♥sã♦ ❢r❛❝t❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
❊s♣❛ç♦ ❞❡ ❢❛s❡✴❡st❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
❊①♣♦❡♥t❡s ❞❡ ▲②❛♣✉♥♦✈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼
❋❛❧s♦s ✈✐③✐♥❤♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽
■♥❢♦r♠❛çã♦ ♠út✉❛ ♠é❞✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽
✶✵
✶ ❖❜❥❡t✐✈♦s ❡ ❏✉st✐✜❝❛t✐✈❛s
❆ ❞✐ss❡rt❛çã♦ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛ ❜✉s❝❛ ❞❡ ❡✈✐❞ê♥❝✐❛s ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♠♣♦♥❡♥t❡ ❞❡t❡r♠✐♥íst✐❝♦ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ♣r❡ç♦s ❞❡ ❝❡rt❛s ❛çõ❡s ♥❡❣♦❝✐❛❞❛s ♥❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ❙ã♦ P❛✉❧♦ ✭❇❖❱❊❙P❆✮ ❡ ❡♠ í♥❞✐❝❡s ❛♠♣❧♦s ❞♦ ♠❡r❝❛❞♦✱ t❛✐s ❝♦♠♦ ♦ ■❜♦✈❡s♣❛ ❡ ■❇❳ ❡✱ ❝♦♠♦ ❝♦♥s❡qüê♥❝✐❛✱ ❞❡t❡r♠✐♥❛r s❡ ❛ ❍✐♣ót❡s❡ ❞❡ ▼❡r❝❛❞♦ ❊✜❝✐❡♥t❡ é ✈á❧✐❞❛ ♣❛r❛ ♦ ♠❡r❝❛❞♦ ❛❝✐♦♥ár✐♦ ❜r❛s✐❧❡✐r♦✳
➱ ♥❡❝❡ssár✐❛ ✉♠❛ rá♣✐❞❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ❍✐♣ót❡s❡ ❞❡ ▼❡r❝❛❞♦ ❊✜❝✐❡♥t❡ ✭❍▼❊✮✱ ✐♥✲ tr♦❞✉③✐❞❛ ♣♦r ❋❛♠❛ ✭✶✾✻✺✮ ♣❛r❛ q✉❡ ❛ ❞✐s❝✉ssã♦ t❡♥❤❛ ❝♦♥t✐♥✉✐❞❛❞❡✳ ❊❧❛ ❛✜r♠❛✱ ❞❡ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛ q✉❡✿
• ▼♦✈✐♠❡♥t♦s s✉❝❡ss✐✈♦s ♥♦s ♣r❡ç♦s ❞❡ ✉♠ ❛t✐✈♦ sã♦ ❡st❛t✐st✐❝❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱
♦✉ s❡❥❛✱ ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ❛çã♦ ❤♦❥❡✱ ♣♦r ❡①❡♠♣❧♦✱ ♥ã♦ ✐♥✢✉❡♥❝✐❛rá ♦ s❡✉ ♣r❡ç♦ ❛♠❛♥❤ã✳ ❖✉tr❛ ❢♦r♠❛ ❞❡ s❡ ❝♦❧♦❝❛r ❡ss❛ ♣r❡♠✐ss❛ é ❛✜r♠❛r q✉❡ ♦s r❡t♦r♥♦s ❞❡ ✉♠ ❛t✐✈♦ sã♦ ♥♦r♠❛❧♠❡♥t❡ ❞✐str✐❜✉í❞♦s✳
• ❊♠ ✉♠ ♠❡r❝❛❞♦ ❡✜❝✐❡♥t❡✱ t♦❞❛s ❛s ✐♥❢♦r♠❛çõ❡s ❞✐s♣♦♥í✈❡✐s sã♦ ✐♥st❛♥t❛♥❡❛♠❡♥t❡
✐♥❝♦r♣♦r❛❞❛s ❛♦ ♣r❡ç♦ ❞❛ ❛çã♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❋❛♠❛✱ três ✧✈❡rsõ❡s✧♣♦ssí✈❡✐s ❞❛ ❍▼❊✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ q✉ã♦ r❡str✐t❛ é ❡ss❛ ❝♦♥❞✐çã♦✿
✕ ❍✐♣ót❡s❡ ❢r❛❝❛✿ t♦❞❛ ✐♥❢♦r♠❛çã♦ ❞✐s♣♦♥í✈❡❧ ❛♦ ♣ú❜❧✐❝♦ é ❝♦♥t❛❜✐❧✐③❛❞❛ ♥♦ ♣r❡ç♦ ❞❛ ❛çã♦✳
✕ ❍✐♣ót❡s❡ s❡♠✐✲❢♦rt❡✿ t♦❞❛ ✐♥❢♦r♠❛çã♦ ❞✐s♣♦♥í✈❡❧ ❛♦ ♣ú❜❧✐❝♦ ❡ ✐♥s✐❞❡rs ❞❛ ❡♠✲ ♣r❡s❛ ❥á s❡ ❡♥❝♦♥tr❛ ♥♦ ♣r❡ç♦ ❞❛ ❛çã♦✳
✕ ❍✐♣ót❡s❡ ❢♦rt❡✿ t♦❞❛ ✐♥❢♦r♠❛çã♦ ❞✐s♣♦♥í✈❡❧ ❛♦ ♣ú❜❧✐❝♦ ❡ ✐♥s✐❞❡rs✱ ❛❧é♠ ❞❡ ✐♥✲ ❢♦r♠❛çõ❡s ❞❡s❝♦♥❤❡❝✐❞❛s ❛ ❛♠❜♦s✱ é ✐♥st❛♥t❛♥❡❛♠❡♥t❡ ✐♥❝♦r♣♦r❛❞❛ ♥♦ ♣r❡ç♦ ❞❛ ❛çã♦✳
✶ ❖❜❥❡t✐✈♦s ❡ ❏✉st✐✜❝❛t✐✈❛s ✶✶
❞❡ ❢♦r♠❛ q✉❡✱ ❡♠ ✉♠❛ sér✐❡ ❞❡ t❡♠♣♦ xt✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❞❡t❡r♠✐♥íst✐❝❛f t❛❧ q✉❡
xt=f(xt−1, xt−2, xt−3, . . . , et−1, et−2, et−3) +et, ✭✶✳✶✮
♦♥❞❡ et sã♦ ❡rr♦s ♦❜s❡r✈❛❞♦s ❝♦♠ ♠é❞✐❛ ③❡r♦✱ ♠❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡
✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞♦s ✭✐✳✐✳❞✳✮ ❞✐str✐❜✉í❞♦s✳
❖r❛✱ ✉♠ s✐st❡♠❛ q✉❡ ❛♣r❡s❡♥t❡ ✉♠ ❝♦♠♣♦♥❡♥t❡ ❞❡t❡r♠✐♥íst✐❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ ❛s✲ s✉♠✐r♠♦s ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ❛çã♦✱ t❡rá ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ❝❛r❛❝t❡ríst✐❝❛ ♦ ❢❛t♦ ❞❡ q✉❡ s❡✉ ♣r❡ç♦ ❤♦❥❡ ✐♥✢✉❡♥❝✐❛rá ♦ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❢✉t✉r♦✱ ♠❡s♠♦ q✉❡ t❛❧ r❡❧❛çã♦ ❡♥❝♦♥tr❡✲s❡ ✧♠❛s✲ ❝❛r❛❞❛✧♣♦r s✐♥❛✐s ❡st♦❝ást✐❝♦s✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛ é ✉♠❛ ❝❧❛r❛ ✈✐♦❧❛çã♦ ❞❛ ❍▼❊✱ t❛❧ ❝♦♠♦ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❋❛♠❛✳
❆ss✐♠✱ ✉♠❛ ♠❛♥❡✐r❛ ♣♦t❡♥❝✐❛❧ ❞❡ ❞❡t❡r♠✐♥❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❡✈✐❞ê♥❝✐❛s ❞❛ ❡✜❝✐ê♥❝✐❛ ❞❡ ✉♠ ♠❡r❝❛❞♦ é s❡r✐❛ ❛tr❛✈és ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♣♦ssí✈❡✐s ❝♦♠♣♦♥❡♥t❡s ❞❡t❡r♠í♥✐st✐❝♦s ♥♦ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦✳ ❚r❛❞✐❝✐♦♥❛❧♠❡♥t❡✱ ✐ss♦ é ❢❡✐t♦ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ té❝♥✐❝❛s ❡❝♦♥♦♠étr✐❝❛s t❛✐s ❝♦♠♦ ❛ ❛♥á❧✐s❡ ❞❡ r❡s❝❛❧♦♥❛♠❡♥t♦ ♦✉ ❘✴❙✱ ❝♦♠♦ ✐♥✐❝✐❛❧♠❡♥t❡ r❡❛❧✐③❛❞♦ ♣♦r ▲♦ ✭✶✾✾✶✮ ❡ r❡♣❧✐❝❛❞♦ ♣♦r ▲✐♠❛ ❡ ❳✐❛♦ ✭✷✵✵✹✮✱ ❛ ❛♥á❧✐s❡ ❞❛ ❢✉♥çã♦ ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦ ❞❡ ✉♠❛ sér✐❡✱ ♦✉ ❛✐♥❞❛✱ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s ❧✐♥❡❛r❡s✱ ❝♦♠♦ ❛♣r❡s❡♥✲ t❛❞♦ ♣♦r ▲❡❛❧ ❡ ❘✐❜❡✐r♦ ✭✷✵✵✵✮✱ ❈❛✈❛❧❝❛♥t✐ ✭✷✵✵✸✮✳ P♦ré♠✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ ♦❜s❡r✈❛çõ❡s ♣❛ss❛❞❛s ❥á é ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❤á ♠✉✐t♦ t❡♠♣♦ ♥❛ ❛♥á❧✐s❡ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♥ã♦✲❧✐♥❡❛r❡s✱ ❝♦♠ té❝♥✐❝❛s ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❞❡s❡♥✈♦❧✈✐❞❛s ❛ ♣❛rt✐r ❞❛ t❡♦r✐❛ ❞♦ ❝❛♦s✱ ❣❡r❛❧♠❡♥t❡ ♦r✐❣✐♥❛❞❛s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❛ ❋ís✐❝❛✱ té❝♥✐❝❛s ❡ss❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✳
❏á ❢♦✐ ♠❡♥❝✐♦♥❛❞♦ q✉❡ ❛ ❍▼❊ ❛ss✉♠❡ ❝♦♠♦ ✉♠❛ ❞❛s s✉❛s ❤✐♣ót❡s❡s ♠❛✐s ❢✉♥❞❛♠❡♥✲ t❛✐s s✉❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❞♦s r❡t♦r♥♦s ❞♦s ❛t✐✈♦s✱ ♦✉ s❡❥❛✱ ♦s ❡✈❡♥t♦s ♠❡❞✐❞♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♥ã♦ ✐♥✢✉❡♥❝✐❛♥❞♦ ✉♥s ❛♦s ♦✉tr♦s ❡ ❝♦♠ ❛ ♠❡s♠❛ ♣r♦❜❛❜✐❧✲ ✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛✳ ❊ss❛ ♣r❡♠✐ss❛ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛✱ ♦✉ ♥♦r♠❛❧✐❞❛❞❡ ❞♦s r❡t♦r♥♦s✱ é ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♠❡♥t❡ ♥♦ ❡st✉❞♦ ❛❝❛❞ê♠✐❝♦ ❞❡ ✜♥❛♥ç❛s✱ ✉♠❛ ✈❡③ q✉❡ ❡❧❛ ♣❡r♠✐t❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❛st♦ ❛rs❡♥❛❧ ❡st❛tíst✐❝♦ s♦❜r❡ ♦s ♣r♦❜❧❡♠❛s ❡st✉❞❛❞♦s✳
■♥✐❝✐❛❧♠❡♥t❡✱ ❡ss❛ ♣r❡♠✐ss❛ ❡r❛ ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ❛❧❣♦ ❡①tr❡♠❛♠❡♥t❡ só❧✐❞♦✱ t❛♥t♦ ❡♠ t❡r♠♦s t❡ór✐❝♦s ❝♦♠♦ ❡♠♣ír✐❝♦s✳ ❚❛♥t♦ q✉❡ ❞✐✈❡rs♦s ❛✉t♦r❡s ✐♠♣♦rt❛♥t❡s ❝❤❡❣❛r❛♠ ❛ ❞❡❝❧❛r❛r q✉❡
✧✳ ✳ ✳ ♥ã♦ ❤á ❡✈✐❞ê♥❝✐❛ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛s s✐❣♥✐✜❝❛t✐✈❛s✶✱ t❛♥t♦ ❞❡ ✉♠ ♣♦♥t♦
❞❡ ✈✐st❛ ❞❡ ✐♥✈❡st✐♠❡♥t♦ ♦✉ ❡st❛tís❝♦✳ ✭❋❆▼❆✱ ✶✾✻✺✮
✶❋❛♠❛ ❞❡t❡❝t♦✉ ❡✈✐❞ê♥❝✐❛s ❞❡ ✉♠❛ ❛✉t♦❝♦rr❡❧❛çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❞❡ ❝❡r❝❛ ❞❡ ❛♣❡♥❛s ✵✱✵✸ ❡♠ s❡✉
✶ ❖❜❥❡t✐✈♦s ❡ ❏✉st✐✜❝❛t✐✈❛s ✶✷
♦✉ ❛✐♥❞❛✿
✧✳ ✳ ✳ ♥ã♦ ❤á ♦✉tr❛ ♣r♦♣♦s✐çã♦ ❡♠ t♦❞❛ ❛ ❊❝♦♥♦♠✐❛ q✉❡ ♣♦ss✉❛ t❛♠❛♥❤❛ ❡✈✐❞ê♥❝✐❛ ❡♠♣ír✐❝❛ ❛ s❡✉ ❢❛✈♦r q✉❛♥t♦ ❛ ❍✐♣ót❡s❡ ❞❡ ▼❡r❝❛❞♦s ❊✜❝✐❡♥t❡s✳ ✭❏❊◆❙❊◆✱ ✶✾✼✽✱ ♣✳ ✾✻✮✧
P♦ré♠✱ ❋❛♠❛ ✭✶✾✼✵✮ ❥á ❝♦♠❡ç❛ ❛ ❛❝✉♠✉❧❛r r❡s✉❧t❛❞♦s ❝♦♥trár✐♦s à ❤✐♣ót❡s❡ ❞❡ ✐♥❞❡✲ ♣❡♥❞ê♥❝✐❛ ❞♦s r❡t♦r♥♦s ❡✱ ❛ ♣❛rt✐r ❞♦ tr❛❜❛❧❤♦ ❞❡ ❏❡♥s❡♥✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡✈✐❞ê♥❝✐❛s ❝♦♥✲ trár✐❛s ♣✉❜❧✐❝❛❞❛s ♣♦r ❞✐✈❡rs♦s tr❛❜❛❧❤♦s✱ ❞❡st❛❝❛♥❞♦✲s❡ ♦s ❞❡ ❲❛tts ✭✶✾✼✽✮✱ ❇❛❧❧ ✭✶✾✼✽✮✱ ❑❡✐♠ ❡ ❙t❛✉♠❜❛✉❣❤ ✭✶✾✽✻✮✱ ❋r❡♥❝❤ ❡ ❘♦❧❧ ✭✶✾✽✻✮✱ ▲♦ ❡ ▼❛❝❑✐♥❧❛② ✭✶✾✽✽✮✱ ❋❛♠❛ ❡ ❋r❡♥❝❤ ✭✶✾✽✽✮✱ ▲♦ ❡ ▼❛❝❑✐♥❧❛② ✭✶✾✾✵✮✱ ❋❛♠❛ ✭✶✾✾✶✮ ♠♦str♦✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ✐♥❛❞✐á✈❡❧ ❞❡ ✉♠❛ ❛♠♣❧❛ r❡✈✐sã♦ ❞❡ss❛ ♣r❡♠✐ss❛✳ ▼✐♥❛r❞✐ ✭✷✵✵✷✮✱ ♣r♦❝✉r♦✉ ❡ ❡♥❝♦♥tr♦✉✱ ❡✈✐❞ê♥❝✐❛s ♥❡ss❡ s❡♥t✐❞♦ ♥♦ ♠❡r❝❛❞♦ ❜r❛s✐❧❡✐r♦✳
❈♦♠♦ ♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦ ♥ã♦ ❛t❡♥❞❡ ❛ ❡ss❛ r❡str✐çã♦ ❞❡ ♥♦r♠❛❧✐❞❛❞❡ ✕ ♣r♦❜❧❡♠❛ ♠✉✐t♦ ♣r♦✈❛✈❡❧♠❡♥t❡ ✐❞❡♥t✐✜❝❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ▼❛♥❞❡❧❜r♦t ✭✶✾✻✸❜✱ ✶✾✻✸❛✮✱ ❋❛♠❛ ✭✶✾✻✺✮✱ ♠❛s só ♥♦t❛❞❛♠❡♥t❡ ♣♦♣✉❧❛r✐③❛❞♦ ♣❡❧♦ ❛rt✐❣♦ ❞❡ ▲♦ ❡ ▼❛❝❑✐♥❧❛② ✭✶✾✽✽✮ ✕ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ❞✐✈❡rs♦s ❛❥✉st❡s q✉❡ ❝r✐❛♠ ❡str✉t✉r❛s ❡st❛tíst✐❝❛s q✉❡✱ ♠❡s♠♦ ♥ã♦ s❡♥❞♦ ♥♦r♠❛✐s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ sã♦ ♣ró①✐♠❛s ♦ ❜❛st❛♥t❡ ♣❛r❛ ♣❡r♠✐t✐r ♦ ✉s♦ ❞♦ ❢❡rr❛♠❡♥t❛❧ ❡st❛tíst✐❝♦ tr❛❞✐❝✐♦♥❛❧✱ ♠❡s♠♦ q✉❡ ❝♦♠ ❛❧❣✉♠❛s ❛❞❛♣t❛çõ❡s✳ ▼❛s ❡ss❛ ❧ó❣✐❝❛✱ ❛❧✐❛❞❛ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❛♥á❧✐s❡ ❡st❛tíst✐❝❛ ❞♦s ♠❡r❝❛❞♦ ✈❡✐♦ ❛♥t❡s ❞♦ s✉r❣✐♠❡♥t♦ ❞❛ ❍▼❊✱ t❡♠ ♥♦s ❧❡✈❛❞♦ ❛ ✉♠ ♣r♦❝❡ss♦ ❞❡ r❛❝✐♦♥❛❧✐③❛çã♦ q✉❡ t❡♥t❛ ❥✉st✐✜❝❛r ♦ ✉s♦ ❞❡ té❝♥✐❝❛s ❡st❛tíst✐❝❛s ❡♠ ✉♠ ❛♠❜✐❡♥t❡ ✐♠♣ró♣r✐♦✱ ♠❛s q✉❡ t❡♠ s✐❞♦ tã♦ ✉s❛❞❛s ♥♦s ú❧t✐♠♦s ✸✵ ❛♥♦s ❞❡ ♣❡sq✉✐s❛ ❛❝❛❞ê♠✐❝❛ q✉❡ ❡❧❛s s❡ t♦r♥❛r❛♠ q✉❛s❡ ✉♠ ✈í❝✐♦✱ s❡♥❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s ❞✐❢í❝✐❧ ❞❡✐①á✲❧❛s ❞❡ ❧❛❞♦✳
❈♦♠❜✐♥❛❞♦ ❛ ❡ss❡ ♣r♦❜❧❡♠❛✱ ❛✐♥❞❛ t❡♠♦s ❛ r❡❧❛t✐✈❛♠❡♥t❡ r❡❝❡♥t❡ ❛t❡♥çã♦ ❞❛❞❛ às ❞✐str✐❜✉✐çõ❡s ❡stá✈❡✐s✱ ❞❡s❡♥✈♦❧✈✐❞❛s ♣♦r ▲é✈② ✭✶✾✷✺✮✱ q✉❡ tê♠ ❝♦♥s✐st❡♥t❡♠❡♥t❡ ❞❡♠♦♥✲ str❛❞♦ ✉♠❛ ♠❡❧❤♦r ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❞❡s❝r✐çã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s r❡t♦r♥♦s ❞♦s ♣r❡ç♦s ❞❡ ❛çõ❡s ▲❡❛❧ ❡ ❘✐❜❡✐r♦ ✭✷✵✵✵✮✱ ❘♦♠❛♥♦✈s❦② ✭✷✵✵✵✮✱ ▼❛♥t❡❣♥❛ ❡ ❙t❛♥❧❡② ✭✶✾✾✹✮✱ ●♦♣✐❦r✲ ✐s❤♥❛♥ ❡t ❛❧✳ ✭✶✾✾✾✮✱ ❱✐s✇❛♥❛t❤❛♥ ❡t ❛❧✳ ✭✷✵✵✷✮✱ ❈❛✈❛❧❝❛♥t✐ ✭✷✵✵✸✮✱ ✉♠❛ ✈❡③ q✉❡ ❡❧❛s ❛♣r❡s❡♥t❛♠ ❛s ❝♦♥❤❡❝✐❞❛s ✧❝❛✉❞❛s ❣♦r❞❛s✧❡♠♣✐r✐❝❛♠❡♥t❡ ♦❜s❡r✈❛❞❛s✱ ♦✉ s❡❥❛✱ ✉♠❛ ❝✉r✲ t♦s❡ ♠❛✐s ♣ró①✐♠❛ à r❡❛❧✐❞❛❞❡✳ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❢r❡qüê♥❝✐❛ ❡①❝❡ss✐✈❛ ❞❡ ❡✈❡♥t♦s ❡①tr❡♠♦s✱ ✉♠❛ ❝rít✐❝❛ ❜❛st❛♥t❡ ❝♦♠✉♠ à ❛♣❧✐❝❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❡stá✈❡✐s✱ t❛♠❜é♠ ❢♦✐ r❡s♦❧✈✐❞❛ ▼❛♥t❡❣♥❛ ❡ ❙t❛♥❧❡② ✭✶✾✾✹✮✱ ♦ q✉❡ t♦r♥❛ ❛✐♥❞❛ ♠❛✐s ✈✐sí✈❡❧ ❛ ♥ã♦✲♥♦r♠❛❧✐❞❛❞❡ ❞❛s sér✐❡s ❞❡ t❡♠♣♦ ✜♥❛♥❝❡✐r❛s✳
✶ ❖❜❥❡t✐✈♦s ❡ ❏✉st✐✜❝❛t✐✈❛s ✶✸
❥á ❧❡✈❛♠ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❢❛t♦ ❞❡ q✉❡ ♦s ❞❛❞♦s ❡st✉❞❛❞♦s ♥ã♦ ❛t❡♥❞❡♠ ❛♦ r❡q✉✐s✐t♦ ❞❡ ♥♦r♠❛❧✐❞❛❞❡ ❡ q✉❡ s❡✉s ❡✈❡♥t♦s ♥ã♦ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tr❡ s✐✳ ❊ss❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❥á é ❧❛r❣❛♠❡♥t❡ ❞❡♠♦♥str❛❞❛ ❡♠ ✐♥ú♠❡r♦s ❡st✉❞♦s ✐♥t❡r♥❛❝✐♦♥❛✐s ❙❝❤❡✐♥❦♠❛♥ ❡ ▲❡❜❛r♦♥ ✭✶✾✽✾✮✱ ❍s✐❡❤ ✭✶✾✾✶✮✱ ▲❡❜❛r♦♥ ✭✶✾✾✹✮✱ ❆❜❤②❛♥❦❛r✱ ❈♦♣❡❧❛♥❞ ❡ ❲♦♥❣ ✭✶✾✾✺✮✱ ▼❛♥t❡❣♥❛ ❡ ❙t❛♥❧❡② ✭✶✾✾✺✮✱ ❙❝❤✐tt❡♥❦♦♣❢✱ ❉♦r✛♥❡r ❡ ❉♦❝❦♥❡r ✭✷✵✵✶✮✱ ●❛rr❡tt ❡ ❚❛②❧♦r ✭✷✵✵✶✮ ❡ ❞✐✈❡r✲ s❛s ♣✉❜❧✐❝❛çõ❡s ♥♦ ❇r❛s✐❧ ❇r✉♥❞♦ ✭✶✾✾✽✮✱ ❋r❛♥❝✐s❝♦✱ P❛✐✈❛ ❡ ❘♦s❡♥❢❡❧❞ ✭✷✵✵✷✮✱ ❈❛✈❛❧❝❛♥t✐ ✭✷✵✵✸✮ ✳ P♦ré♠✱ ❡♠ ✉♠ ❛rt✐❣♦ r❡❝❡♥t❡♠❡♥t❡ ♣✉❜❧✐❝❛❞♦✱ ❙♠❛❧❧ ❡ ❚s❡ ✭✷✵✵✸✮ ♠♦str❛r❛♠ q✉❡✱ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞♦s✱ t❛❧ ❛♥á❧✐s❡ é ❢❡✐t❛ ❞❡ ❢♦r♠❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ rí❣✐❞❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❛♣❡♥❛s ❞✉❛s ❝❧❛ss✐✜❝❛çõ❡s ❡r❛♠ ♣♦ssí✈❡✐s às sér✐❡s ❛♥❛❧✐s❛❞❛s✿ ♦✉ ❛ sér✐❡ ❛♣r❡s❡♥✲ t❛✈❛ ❝♦♠♣♦rt❛♠❡♥t♦ ❝❛ót✐❝♦✱ ♦✉ ♥ã♦✳ ▼❛s✱ ❝♦♠♦ é s❛❜✐❞♦ ❞❛ ❡①♣❡r✐ê♥❝✐❛ ❡♠♣ír✐❝❛ ❡♠ ❋✐♥❛♥ç❛s ❡ ❊❝♦♥♦♠✐❛✱ sér✐❡s ❞❡ t❡♠♣♦ ✜♥❛♥❝❡✐r❛s ❛♣r❡s❡♥t❛♠✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❝♦♠♣♦✲ ♥❡♥t❡s ❡st♦❝ást✐❝♦s ✭❛❧❡❛tór✐♦s✮ ❡ ❝❛ót✐❝♦s ✭❞❡t❡r♠✐♥íst✐❝♦s✮✳ ❉❡ss❛ ❢♦r♠❛✱ ❡ss❡ tr❛❜❛❧❤♦ t❡♥t❛rá ✐♥tr♦❞✉③✐r ✉♠❛ ❡str✉t✉r❛ ❞❡ ❛♥á❧✐s❡ ♠❛✐s ✢❡①í✈❡❧ ❡ ❝♦♠ té❝♥✐❝❛s ✉♠ ♣♦✉❝♦ ♠❛✐s ❛t✉❛❧✐③❛❞❛s ❞♦ q✉❡ ❛rt✐❣♦s ❥á ♣✉❜❧✐❝❛❞♦s ❛té ♦ ♠♦♠❡♥t♦ ♥♦ ❇r❛s✐❧✳
P♦r ❡①❡♠♣❧♦✱ ❇r✉♥❞♦ ✭✶✾✾✽✮ ❞❡s❝r❡✈❡✉ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ❞❛s ❡st❛tíst✐❝❛s ❞❡ ✉s♦ ♠❛✐s ❛♠♣❧♦ ♥❛ ❞❡t❡❝çã♦ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ♥ã♦✲❧✐♥❡❛r✱ ❛ ❡st❛tís❝❛ ❇❉❙✱ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❇r♦❝❦ ❡t ❛❧✳ ✭✶✾✾✺✮✱ q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ❛s ♦❜s❡r✈❛çõ❡s ❛♥❛❧✐s❛❞❛s tê♠ ❝♦♠♦ ♣r♦❝❡ss♦ ❣❡r❛❞♦r ✉♠ s✐st❡♠❛ ❝❛ót✐❝♦✳ ❚❛❧ ❡st❛tíst✐❝❛ r❡♣r❡s❡♥t❛ ✉♠ t❡st❡ r✐❣♦r♦s♦ ♣❛r❛ ❝❛♦s ❞❡t❡r♠✐♥íst✐❝♦✱ ♣♦ré♠✱ ♥ã♦ s❡ s❛❜✐❛ ♥❛ ❞❛t❛ ❞❡ ♣✉❜❧✐❝❛çã♦ ❞♦ tr❛❜❛❧❤♦✱ ♦s ❡❢❡✐t♦s ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✜❧tr♦s ❧✐♥❡❛r❡s s♦❜r❡ ❡ss❛ ❡st❛tíst✐❝❛✳ ❊ss❡s ❡❢❡✐t♦s s❡rã♦ ❞✐s❝✉t✐❞♦s ♥❛ s❡çã♦ ✭✷✳✶✮✳
❖✉tr♦s ❛✉t♦r❡s ❜r❛s✐❧❡✐r♦s✱ ♥♦t❛❞❛♠❡♥t❡ ❋r❛♥❝✐s❝♦✱ P❛✐✈❛ ❡ ❘♦s❡♥❢❡❧❞ ✭✷✵✵✷✮✱ ❘♦❝❤♠❛♥ ✭✷✵✵✷✮✱ t❛♠❜é♠ ❥á ❛♣❧✐❝❛r❛♠ té❝♥✐❝❛s ♦r✐❣✐♥❛❞❛s ❞❛ ❋ís✐❝❛ s♦❜r❡ sér✐❡s t❡♠♣♦r❛✐s ❡❝♦♥ô♠✐✲ ❝❛s ❡ ✜♥❛♥❝❡✐r❛s✱ ♠♦str❛♥❞♦ q✉❡ ♦ ✉s♦ ❞♦ ❢❡rr❛♠❡♥t❛❧ ♣r♦✈❡♥✐❡♥t❡ ❞♦ ❝❛♦s ❞❡t❡r♠✐♥íst✐❝♦ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ❛♣❧✐❝❛❞♦ ❛♦s ❞❛❞♦s ❜r❛s✐❧❡✐r♦s ❝♦♠ r❡s✉❧t❛❞♦s ❞✐❣♥♦s ❞❡ ♠❛✐♦r ❡st✉❞♦ ❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦✳
❊♠ t❡r♠♦s ❞❡ s✐♠✐❧❛r✐❞❛❞❡✱ ❛ ♠❡t♦❞♦❧♦❣✐❛ ✉s❛❞❛ ♣♦r ❋r❛♥❝✐s❝♦✱ P❛✐✈❛ ❡ ❘♦s❡♥❢❡❧❞ ✭✷✵✵✷✮ s❡ ❛♣r♦①✐♠❛ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ♥♦ q✉❡ s❡ r❡❢❡r❡ à r❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ♠❛s ❝♦♠ ❞✐❢❡r❡♥ç❛s ✐♠♣♦rt❛♥t❡s✳ Pr✐♠❡✐r♦✱ ❝❛❧❝✉❧❛♠♦s ❛ ❞❡❢❛s❛❣❡♠ ❞❡ t❡♠♣♦ ♣❛r❛ ❛ r❡❝♦♥str✉çã♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❢❛s❡ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❛ ✐♥❢♦r♠❛çã♦ ♠út✉❛ ♠é❞✐❛ s✉❣❡r✐❞♦
♣♦r ❆❜❛r❜❛♥❡❧ ✭✶✾✾✺✮✱ ❛♦ ✐♥✈és ❞❡ ❛ss✉♠✐r♠♦s T = 1✳ ❙❡❣✉♥❞♦✱ ❞❡✜♥✐♠♦s ♦ ♥ú♠❡r♦ ❞❡
✶✳✶ ❈♦♥❝❡✐t♦s ❋✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s ✶✹
✭✶✾✾✸✮✱ ❑❛♥t③ ✭✶✾✾✹✮ q✉❡✱ ❛♦ ❝♦♥trár✐♦ ❞❛q✉❡❧❡ ♦r✐❣✐♥❛❧♠❡♥t❡ ♣r♦♣♦st♦ ♣♦r ❲♦❧❢ ❡t ❛❧✳ ✭✶✾✽✺✮ ❡ ❡♠♣r❡❣❛❞♦ ♣♦r ❋r❛♥❝✐s❝♦✱ P❛✐✈❛ ❡ ❘♦s❡♥❢❡❧❞ ✭✷✵✵✷✮✱ ♥ã♦ ❛ss✉♠❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡t❡r♠✐♥✐s♠♦ ♥❛ sér✐❡✳
✶✳✶ ❈♦♥❝❡✐t♦s ❋✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s
P❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✱ t❡♥t❛r❡♠♦s ❛♣r❡s❡♥t❛r ❞❡ ♠❛♥❡✐r❛ ✐♥t✉✲ ✐t✐✈❛ ❡✱ ❝♦♥s❡qü❡♥t❡♠❡♥t❡✱ ✉♠ t❛♥t♦ ✐♥❢♦r♠❛❧ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s✳ ❯♠❛ ✈❡③ q✉❡ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ❢♦❝♦ sér✐❡s ❞❡ t❡♠♣♦ ✜♥❛♥❝❡✐r♦s✱ ❡✈✐t❛r❡✲ ♠♦s ❞✐s❝✉ssõ❡s ❞❡t❛❧❤❛❞❛s ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❡♥✈♦❧✈✐❞♦s✱ ❢♦❝❛♥❞♦ ❛ ❞✐s❝✉ssã♦ ♥♦s ♣❛ss♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡ ❝♦♥s❡qüê♥❝✐❛s ❞❛s ♠❡s♠❛s ❛♦ tr❛❜❛❧❤♦✳
✶✳✶✳✶ ❈❛♦s ❡ ❈♦♠♣♦rt❛♠❡♥t♦ ❈❛ót✐❝♦
❋♦r♠❛❧♠❡♥t❡✱ ♥ã♦ ❤á ❞❡✜♥✐çã♦ ♣❛r❛ ♦ t❡r♠♦ ❝❛♦s✱ ♣♦ré♠✱ ❞✐✈❡rs♦s ❛✉t♦r❡s✱ t❛✐s ❝♦♠♦ ❙tr♦❣❛t③ ✭✶✾✾✹✮✱ ❆❜❛r❜❛♥❡❧ ✭✶✾✾✺✮✱ ❑❛♥t③ ❡ ❙❝❤r❡✐❜❡r ✭✷✵✵✹✮ ♠❡♥❝✐♦♥❛♠ três ❝❛r❛❝t❡rís✲ t✐❝❛s q✉❡ ❛❝❛❜❛♠ ❝♦♠♣♦♥❞♦ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❈❛♦s é ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♣❡r✐ó❞✐❝♦ ❞❡ ❧♦♥❣♦ ♣r❛③♦ ❡♠ ✉♠ s✐st❡♠❛ ❞❡t❡r♠✐♥íst✐❝♦ q✉❡ ❡①✐❜❡ s❡♥s✐❜✐❧✐❞❛❞❡ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✳
✐ ❆ ❡①♣r❡ssã♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♣❡r✐ó❞✐❝♦ ❞❡ ❧♦♥❣♦ ♣r❛③♦ r❡❢❡r❡✲s❡ às tr❛❥❡tór✐❛s q✉❡ ♥ã♦
❝♦♥✈❡r❣❡♠ ❛ ♣♦♥t♦s ✜①♦s✱ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ♦✉ q✉❛s❡✲♣❡r✐ó❞✐❝❛s q✉❛♥t♦ t → ∞✳
◆ã♦ s❡r✐❛ ❡①❛❣❡r❛❞♦ ❛✜r♠❛r q✉❡ t❛✐s tr❛❥❡tór✐❛s ♦❝♦rr❡r✐❛♠ ❝♦♠ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ❞❛❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❛❧❡❛tór✐❛s✳
✐✐ ❖ t❡r♠♦ ❞❡t❡r♠✐♥íst✐❝♦ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♦ s✐st❡♠❛ ♥ã♦ é ❛❧✐♠❡♥t❛❞♦ ❝♦♠ ❞❛❞♦s ♦✉ ♣❛râ♠❡tr♦s ❛❧❡❛tór✐♦s✱ ♦✉ s❡❥❛✱ ✧r✉í❞♦s✧✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ✐rr❡❣✉❧❛r ♦r✐❣✐♥❛✲ s❡ ❞❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ✐♥trí♥s❡❝❛ ❛♦ ♣ró♣r✐♦ s✐st❡♠❛✳
✐✐✐ ❆ s❡♥s✐❜✐❧✐❞❛❞❡ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s s✐❣♥✐✜❝❛ q✉❡ tr❛❥❡tór✐❛s ✐♥✐❝✐❛❧♠❡♥t❡ ♣ró①✐♠❛s ✉♠❛s ❞❛s ♦✉tr❛s s❡ ❛❢❛st❛♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ t❛①❛s ❡①♣♦♥❡♥❝✐❛✐s✳
✶✳✶✳✷ ❆tr❛t♦r❡s ❊str❛♥❤♦s
✶✳✶ ❈♦♥❝❡✐t♦s ❋✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❞♦ ❈❛♦s ✶✺
s✐♠♣❧❡s✳ ❆ss✐♠✱ ❡♥q✉❛♥t♦ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✱ ❞❛❞❛ ♣♦r ❙tr♦❣❛t③ ✭✶✾✾✹✮ ❡ ❝♦♠♣❧❡♠❡♥t❛❞❛ ♣♦r ❙♣r♦tt ✭✷✵✵✸✮ ♣♦ss❛ s❡r ❝♦♥s✐❞❡r❛❞❛ ✉♠ t❛♥t♦ ✐♥❢♦r♠❛❧✱ ❡❧❛ ❝♦♥té♠ ❛s ♣r✐♥❝✐♣❛✐s
❝❛r❛❝t❡ríst✐❝❛s ❞❡ ✉♠ ❛tr❛t♦r✳ ❙❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦A❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✐ A é ✉♠ ❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡✿ q✉❛❧q✉❡r tr❛❥❡tór✐❛x(t) q✉❡ ❝♦♠❡❝❡ ❡♠A✱ ♣❡r♠❛♥❡❝❡rá
❡♠ A ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳
✐✐ A ❛tr❛✐ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦
U ❝♦♥t❡♥❞♦ A t❛❧ q✉❡ s❡ x(0) ∈ U✱ ❡♥tã♦ ❛ ❞✐stâ♥❝✐❛ ❞❡ x(t) ♣❛r❛ A t❡♥❞❡rá ❛ ✵
q✉❛♥t♦ t → ∞✳ ❆ss✐♠✱ A ❛tr❛✐rá t♦❞❛s ❛s tr❛❥❡tór✐❛s q✉❡ ❝♦♠❡❝❡♠ s✉✜❝✐❡♥t❡♠❡♥t❡
♣❡rt♦ ❞❡❧❡✳
✐✐✐ A é ♠í♥✐♠♦✿ ♥ã♦ ❤á ♥❡♥❤✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ A q✉❡ s❛t✐s❢❛ç❛ ❛s ❝♦♥❞✐çõ❡s ✭✐✮ ❡ ✭✐✐✮✱
♦✉ s❡❥❛✱ ❡❧❡ é ❡r❣ó❞✐❝♦✳
❆❧é♠ ❞❡ss❛s ❝❛r❛❝t❡ríst✐❝❛s ❜ás✐❝❛s✱ ✉♠ ❛tr❛t♦r ❡str❛♥❤♦ ❡①✐❜❡ ❝♦♥❞✐çõ❡s ❛❞✐❝✐♦♥❛✐s
✐ ❊❧❡ é ✉♠ ❢r❛❝t❛❧✱ ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s ❝♦♠ ❡str✉t✉r❛s ❛✉t♦✲s✐♠✐❧❛r❡s ❡♠ q✉❛❧q✉❡r ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ❡✱ ❣❡r❛❧♠❡♥t❡✱ ❝♦♠ ❞✐♠❡♥sõ❡s ♥ã♦✲✐♥t❡✐r❛s✳
✶✻
✷ ❍✐♣ót❡s❡s ❡ ▼❡t♦❞♦❧♦❣✐❛
❆♦ ❝♦♥trár✐♦ ❞♦ q✉❡ t❡♠ s✐❞♦ ❢❡✐t♦ ❡♠ t❡r♠♦s ❞❡ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❝❛ót✐❝♦ ❞❡t❡r♠✐♥íst✐❝♦ ❡♠ sér✐❡s ❞❡ t❡♠♣♦ ✜♥❛♥❝❡✐r❛s✱ ❝♦♠♦ ❢♦✐ ♠❡♥❝✐♦♥❛❞♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❛ ❞✐ss❡rt❛çã♦ ✈❡r✐✜❝❛rá ❛ ✈❛❧✐❞❛❞❡ ❞❡ ✉♠❛ ❤✐♣ót❡s❡ ♠❛✐s ❛❜r❛♥❣❡♥t❡ ❡✱ ❛ss✐♠✱ ♠❛✐s ✢❡①í✈❡❧ ❡ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ r❡❛❧✐❞❛❞❡ ♦❜s❡r✈❛❞❛ ♥❛ ❛♥á❧✐s❡ ❞❡ss❡ t✐♣♦ ❞❡ sér✐❡ ❞❡ t❡♠♣♦✳
❍✐♣ót❡s❡ ✶✿ ◆ã♦ ❤á ❡✈✐❞ê♥❝✐❛ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♠♣♦♥❡♥t❡ ❞❡t❡r♠✐♥íst✐❝♦ ♥❛
sér✐❡ ❞❡ t❡♠♣♦ ❛♥❛❧✐s❛❞❛✳
❆ss✐♠✱ ❝❧❛ss✐✜❝❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ sér✐❡ ❞❡ t❡♠♣♦ ❝♦♠♦ ❝♦♥t❡♥❞♦ ✉♠ ❝♦♠♣♦♥❡♥t❡ ❞❡t❡r♠✐♥íst✐❝♦ ♦✉ ♥ã♦✳ ➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ t❛❧ ❤✐♣ót❡s❡ ♥ã♦ ❧❡✈❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ♥ã♦✲❧✐♥❡❛r❡s ♥❛ sér✐❡✱ ❢❛t♦ ❝♦♠✉♠ q✉❛♥❞♦ tr❛t❛♠♦s ❞♦ ❝❛s♦ ❞❡ sér✐❡s t❡♠♣♦r❛✐s ✜♥❛♥❝❡✐r❛s✳
❈♦♠ ✐ss♦✱ ❤❛✈❡♥❞♦ ❡✈✐❞ê♥❝✐❛s ❞❡ ❝♦♠♣♦♥❡♥t❡s ❞❡t❡r♠✐♥íst✐❝♦s ♥❛ sér✐❡✱ s❡❣✉❡ ❝♦♠♦ ❝♦♥s❡qüê♥❝✐❛ ♥❛t✉r❛❧ ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ♦ ♣r❡ç♦ ❞♦ ❛t✐✈♦ ❤♦❥❡ ♠❛♥té♠ ❛❧❣✉♠ t✐♣♦ ❞❡ r❡❧❛çã♦ ❝♦♠ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ♣❛ss❛❞♦✳ ❈❛s♦ ❡①✐st❛♠ ✐♥❞✐❝❛çõ❡s ♥❡ss❡ s❡♥t✐❞♦✱ t❡r❡♠♦s ✉♠ ❢♦rt❡ ✐♥❞í❝✐♦ ❞❡ q✉❡ ❛q✉❡❧❡ ❛t✐✈♦ ❡♠ ♣❛rt✐❝✉❧❛r ♥ã♦ s❡r✐❛ ❝♦♠♣❛tí✈❡❧ ❝♦♠✱ ♥❡♠ ❛♦ ♠❡♥♦s✱ ❛ ✈❡rsã♦ ✧❢r❛❝❛✧❞❛ ❍▼❊✳
❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❛ ❤✐♣ót❡s❡ s❡rá ❢❡✐t❛ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ té❝♥✐❝❛s ♦r✐❣✐♥❛❞❛s ❞♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♥ã♦✲❧✐♥❡❛r❡s ❡ q✉❡✱ ❛♣❡s❛r ❞❡ t♦❞❛s ❡❧❛s ♣♦❞❡r❡♠ s❡r ✉t✐❧✐③❛❞❛s
♣❛r❛ ❞❡t❡❝t❛r ❡✈✐❞ê♥❝✐❛s✶❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❝❛ót✐❝♦✱ ❛❧❣✉♠❛s ❞❡❧❛s ❢♦r❛♠ ❡s♣❡❝✐✜❝❛♠❡♥t❡
❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ t❡st❛r ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞❡ ✉♠❛ sér✐❡✱ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ✭♠❛s ♥ã♦ s✉✜❝✐❡♥t❡✮ ♣❛r❛ ♦ ❝❛♦s✳
❖ ♣r✐♠❡✐r♦ ✭❡ t❛❧✈❡③✱ ♦ ♠❛✐s ✐♠♣♦rt❛♥t❡✮ ♣❛ss♦ ♥♦ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ♥❛ r❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ♣❛r❛ q✉❡ ❡♥tã♦ ♣♦ss❛♠♦s ❡st✉❞❛r ❛ ❞✐♥â♠✐❝❛ ❞❛s sér✐❡s✳ ❯♠❛ ❞❡s❝r✐çã♦
✶♥♦ ❝❛s♦ ❞❡ sér✐❡s t❡♠♣♦r❛✐s ✜♥❛♥❝❡✐r❛s✱ ❢r❡qü❡♥t❡♠❡♥t❡ é ✐♠♣♦ssí✈❡❧ ❛✜r♠❛r ❝♦♠ ❝❡rt❡③❛ ❛ ❡①✐stê♥❝✐❛
✷✳✶ ❉❛❞♦s ✶✼
❞❡t❛❧❤❛❞❛ ❞♦s ❝♦♥❝❡✐t♦s ❡ té❝♥✐❝❛s ❡♥✈♦❧✈✐❞❛s é ❞❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✸✳ ❆q✉✐ ❜❛st❛ ❞✐③❡r q✉❡✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❡ss❛ t❛r❡❢❛ é ♦ t❡♦r❡♠❛ ❞❡ ❡♠❜❡❞❞✐♥❣✱ ❛tr✐❜✉í❞♦ ❛ ❚❛❦❡♥s ✭✶✾✽✶✮
❡ q✉❡ ❣❛r❛♥t❡ ❛ tr❛♥s✐çã♦ ❞❡ ♦❜s❡r✈❛çõ❡s ❡s❝❛❧❛r❡ss(k) =s(t0+kτs)❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❢❛s❡
♠✉❧t✐✈❛r✐❛❞♦✳ P❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞❡ss❡ t❡♦r❡♠❛✱ é ♥❡❝❡ssár✐❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❛❞❡q✉❛❞❛ ❞❡
❞♦✐s ♣❛râ♠❡tr♦s✱ ♦ ❧❛❣ ❞❡ t❡♠♣♦ ✭τ✮ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❞✐♠❡♥sõ❡s ✭d✮✳
❯♠❛ ✈❡③ ❢❡✐t♦ ✐ss♦✱ ♣❛ss❛♠♦s à ❜✉s❝❛ ❞❡ ❡✈✐❞ê♥❝✐❛s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡t❡r♠✐♥íst✐❝♦ ♥❛s sér✐❡s✳ ■ss♦ s❡rá r❡❛❧✐③❛❞♦ ❛tr❛✈és ❞♦ ❝á❧❝✉❧♦ ❞❡ ✈❛❧♦r❡s q✉❡ ♥♦s q✉❛♥t✐✜q✉❡♠ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❛ ❞✐♠❡♥sã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡ ♦s ❡①♣♦❡♥t❡s ❞❡ ▲②❛♣✉♥♦✈✱ ❡s❝♦❧❤✐❞❛s ♣♦r ❞❡♣❡♥❞❡♥❞❡r❡♠ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❞♦s ♣r♦❝❡❞✐♠❡♥t♦s ❞❡ ♠❡♥s✉r❛çã♦✱ s✐s✲ t❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡s❝♦❧❤✐❞♦ ❡t❝✳✱ s❡♥❞♦ q✉❛♥t✐❞❛❞❡s ✐♥✈❛r✐❛♥t❡s✳
❈♦♠♦ s❡ t♦r♥♦✉ ♥❡❝❡ssár✐❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ❣r❛♥❞❡ ♣❛rt❡ ❞♦s ❛❧❣♦r✐t♠♦s ✉s❛❞♦s ♥♦ ♣r❡s❡♥t❡✱ ❛ ✜♠ ❞❡ t❡st❛r s✉❛ ✈❛❧✐❞❛❞❡ ❡ ✐♥tr♦❞✉③✐r ♦s ❝♦♥❝❡✐t♦s ✉s❛❞♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡
s❡rã♦ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❞❛❞♦s ❞❡ ❛♥á❧✐s❡ ❛s ♦❜s❡r✈❛çõ❡s ❞♦ ❝♦♠♣♦♥❡♥t❡ x(t) ❞♦ ♠♦❞❡❧♦
❞❡ ▲♦r❡♥③✱ ❞❡s❝r✐t♦ ♣❡❧♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ✭✸✳✾✮✱ s❡♥❞♦ ♦ ♠❡s♠♦ ✉♠ s✐st❡♠❛ ❝❛ót✐❝♦ ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s ❡①t❡♥s✐✈❛♠❡♥t❡ ❡st✉❞❛❞❛s ❡ ❢r❡qü❡♥t❡♠❡♥t❡ ✉s❛❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ ♥♦✈❛s té❝♥✐❝❛s ♥ã♦✲❧✐♥❡❛r❡s✳
❆ss✐♠✱ ❛♣ós ❛ ❛♣r❡s❡♥t❛çã♦ t❡ór✐❝❛ ❡ ✈❛❧✐❞❛çã♦ ❞♦s ❛❧❣♦r✐t♠♦s✱ ♦ ❝❛♣ít✉❧♦ ✺ s❡rá ❞❡❞✲ ✐❝❛❞♦ ❡①❝❧✉s✐✈❛♠❡♥t❡ à ❛♥á❧✐s❡ ❞♦s ❞❛❞♦s ❡♠♣ír✐❝♦s ❝♦♠ ♦ ✉s♦ ❞❡ss❡ ❢❡rr❛♠❡♥t❛❧✱ ♣❡r✲ ♠✐t✐♥❞♦ ✉♠ ❡st✉❞♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ ❞❛s ✈❛♥t❛❣❡♥s ❡ ♣r♦❜❧❡♠❛s ❡s♣❡❝í✜❝♦s r❡❧❛❝✐♦♥❛❞♦s ❛♦ s❡✉ ✉s♦ às sér✐❡s t❡♠♣♦r❛✐s ✜♥❛♥❝❡✐r❛s✳
✷✳✶ ❉❛❞♦s
❆s sér✐❡s t❡♠♣♦r❛✐s ❛♥❛❧✐s❛❞❛s ♥♦ tr❛❜❛❧❤♦ ❢♦r❛♠ ❡s❝♦❧❤✐❞❛s ♣❡❧❛ s✉❛ r❡♣r❡s❡♥t❛t✐✈✐✲ ❞❛❞❡ ♥♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦ ❜r❛s✐❧❡✐r♦✱ ❛ss✐♠ ❝♦♠♦ ♣❡❧❛s s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❧✐q✉✐❞❡③ ✭♦ q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛ ✉♠❛ ♠❛✐♦r q✉❛♥t✐❞❛❞❡ ❞❡ ♦❜s❡r✈❛çõ❡s✮ ❡ ❢r❡qüê♥❝✐❛✳ ❉❡♥tr♦ ❞♦ ♣❡rí♦❞♦ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ♠❛rç♦ ❞❡ ✶✾✾✹ ❡ ❥✉❧❤♦ ❞❡ ✷✵✵✹ ❢♦r❛♠ ❝♦❧❡t❛❞❛s ❛s sér✐❡s ❞✐ár✐❛s ❝♦rr❡✲ s♣♦♥❞❡♥t❡s ❛♦ í♥❞✐❝❡ ■❜♦✈❡s♣❛ ❡ ■❇r❳✱ ♦ ❝â♠❜✐♦ r❡❛❧✲❞ó❧❛r ✭P❚❆❳✮✱ ❛ss✐♠ ❝♦♠♦ ♦s ♣r❡ç♦s ❞❛s ❛çõ❡s ❞❛ P❡tr♦❜rás P◆ ✭P❊❚❘✹✮ ❡ ❊♠❜r❛t❡❧ P❛rt✐❝✐♣❛çõ❡s P◆ ✭❊❇❚P✹✮✳
✷✳✶ ❉❛❞♦s ✶✽
rt = log
pt+1
pt
. ✭✷✳✶✮
♦♥❞❡ pt+1 ❡pt ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ♣r❡ç♦s ❞♦ ❛t✐✈♦ ♥❛s ❞❛t❛s t+ 1❡ t✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❚❛❜❡❧❛ ✷✳✶✳ ❉❡s❝r✐çã♦ r❡s✉♠✐❞❛ ❞❛s sér✐❡s ❞❡ t❡♠♣♦ ❛♥❛❧✐s❛❞❛s✳
❙ér✐❡ ❙í♠❜♦❧♦ Pr✐♠❡✐r❛ ❉❛t❛ Ú❧t✐♠❛ ❉❛t❛ ❖❜s❡r✈❛çõ❡s
■❜♦✈❡s♣❛ ■❇❖❱ ✵✸✴✵✶✴✶✾✾✹ ✶✺✴✶✵✴✷✵✵✹ ✷✳✻✸✹
■❇r❳✲✺✵ ■❇❘❳ ✸✵✴✶✷✴✶✾✾✼ ✶✺✴✶✵✴✷✵✵✹ ✶✳✻✽✹
❈â♠❜✐♦ ❘❡❛❧✲❉ó❧❛r P❚❆❳ ✵✶✴✵✹✴✶✾✾✾ ✶✺✴✶✵✴✷✵✵✹ ✶✳✸✻✾
P❡tr♦❜rás P◆ P❊❚❘✹ ✵✶✴✵✸✴✶✾✾✹ ✶✺✴✶✵✴✷✵✵✹ ✷✳✻✷✾
❊♠❜r❛t❡❧ P❛rt✐❝✐♣❛çõ❡s P◆ ❊❇❚P✹ ✷✶✴✵✾✴✶✾✾✽ ✶✺✴✶✵✴✷✵✵✹ ✶✳✺✵✻
❖ ♥ú♠❡r♦ ❞❡ ♦❜s❡r✈❛çõ❡s ❞❡ ❝❛❞❛ sér✐❡ é ♠♦str❛❞♦ ♥❛ t❛❜❡❧❛ ✭✷✳✶✮✳ ❱❛❧❡ ♥♦t❛r q✉❡✱
♥❛ ♣rát✐❝❛✱ t❡r❡♠♦s n−1 ♦❜s❡r✈❛çõ❡s ❞✐s♣♦♥í✈❡✐s ❞❡✈✐❞♦ ❛♦ ❝á❧❝✉❧♦ ❞♦s r❡t♦r♥♦s ❞✐ár✐♦s✳
P♦❞❡♠♦s ♦❜s❡r✈❛r ❣r❛✜❝❛♠❡♥t❡ ❛ ❡✈♦❧✉çã♦ ❞♦s ♣r❡ç♦s ❞❡ss❡s ❛t✐✈♦s ❛ s❡❣✉✐r✿
❈♦♠♦ ❙❝❤❡✐♥❦♠❛♥ ❡ ▲❡❜❛r♦♥ ✭✶✾✽✾✮ ❝❤❛♠❛♠ ❛ ❛t❡♥çã♦ ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ ❞❛❞♦s ❞✐ár✐♦s ❞❡ ♣r❡ç♦s ❞❡ ❛t✐✈♦s ✜♥❛♥❝❡✐r♦s sã♦ s❡♥sí✈❡✐s ❛ ❢❡♥ô♠❡♥♦s ❞❡ ♠❡r❝❛❞♦✱ t❛✐s ❝♦♠♦ ❡❢❡✐t♦s ❞❡ ✜♠ ❞❡ s❡♠❛♥❛✱ ♣♦r ❡①❡♠♣❧♦✱ s❡♥❞♦ ❛ss✐♠✱ ♠❛✐s r✉✐❞♦s♦s✳ ❉✐❛♥t❡ ❞✐ss♦✱ ❡❧❡s r❡❝♦♠❡♥❞❛♠ ♦ ✉s♦ ❞❡ ❞❛❞♦s s❡♠❛♥❛✐s✱ ♠❛s✱ ❞✐❛♥t❡ ❞♦ ♥ú♠❡r♦ ❜❛st❛♥t❡ ❧✐♠✐t❛❞♦ ❞❡ ♦❜✲ s❡r✈❛çõ❡s✱ s❡❣✉✐r❡♠♦s ❛ ❧✐♥❤❛ ❞❡ ❇r✉♥❞♦ ✭✶✾✾✽✮✱ q✉❡ ✉t✐❧✐③❛ sér✐❡s ❞✐ár✐❛s ❞❡✈✐❞♦ à ♠❡s♠❛ ❧✐♠✐t❛çã♦✳
➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡✱ ❛♦ ❝♦♥trár✐♦ ❞♦ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❢❡✐t♦ ♥❛ ❧✐t❡r❛t✉r❛ ❡st❛tíst✐❝❛ ❡ ❡❝♦♥ô♠✐❝❛✱ ❝♦♠♦ ❛♣❧✐❝❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♣♦r ❇r✉♥❞♦ ✭✶✾✾✽✮ ❡ ♣♦r ❘♦❝❤♠❛♥ ✭✷✵✵✷✮✱ ♥ã♦ ❛♣❧✐❝❛r❡♠♦s ✜❧tr♦s ❧✐♥❡❛r❡s s♦❜r❡ ♦s ❞❛❞♦s ❞❛ ❢♦r♠❛ ❝♦♠♦ ♥♦r♠❛❧♠❡♥t❡ ✐ss♦ é ❢❡✐t♦✱ ♦✉ s❡❥❛✱ ❛♣❧✐❝❛♥❞♦✲s❡ ♦ ❜r❛♥q✉❡❛♠❡♥t♦ t♦t❛❧ ❞❛ sér✐❡✳
✷✳✶ ❉❛❞♦s ✶✾
✷✳✶ ❉❛❞♦s ✷✵
♠❡♥t❡✳
✭✳ ✳ ✳ ✮ ❬♣❛rt✐❝✉❧❛r♠❡♥t❡❪ ♦ ✉s♦ ❞❡ ✜❧tr♦s ❧✐♥❡❛r❡s r❡❧❛❝✐♦♥❛❞♦s ❛♦ ♣r♦❝❡ss♦ ❞❡ ❜r❛♥q✉❡❛♠❡♥t♦ r❛r❛♠❡♥t❡ é ót✐♠♦ ❡✱ ✉s✉❛❧♠❡♥t❡✱ ❞❡❣r❛❞❛ ❛ q✉❛❧✐❞❛❞❡ ❞♦s ❞❛❞♦s✳ ✭✳ ✳ ✳ ✮ ❬❡♠ ❛❧❣✉♥s ❝❛s♦s❪ ❛ ✜❧tr❛❣❡♠ ❧✐♥❡❛r é ✈❛♥t❛❥♦s❛✱ ♠❛s ✉♠ ❜r❛♥q✉❡❛♠❡♥t♦ ❝♦♠♣❧❡t♦ ♥ã♦ é r❡❝♦♠❡♥❞❛❞♦✳ ✭❚❍❊■▲❊❘✱ ✶✾✾✸✱ ♣✳ ✻✮✳
❖ ♣r♦❝❡ss♦ ❡♠ s✐ é ❜❛st❛♥t❡ s✐♠♣❧❡s✳ ❉❛❞❛ ✉♠❛ sér✐❡ t❡♠♣♦r❛❧ xt✱ s❛❜❡♠♦s q✉❡ ♦
♠❡❧❤♦r ♣r❡✈✐s♦r ❧✐♥❡❛r xˆt é ❞❛❞♦ ♣❡❧♦ ♠♦❞❡❧♦
ˆ
xt =a0 +
q
X
k=1
akxt−k ✭✷✳✷✮
❡ q✉❡ ♦s s❡✉s r❡sí❞✉♦s et=xt−xˆt✱ ♦✉ s❡❥❛✱ ❛q✉✐❧♦ q✉❡ ♥ã♦ ❢♦✐ ❝❛♣t✉r❛❞♦ ♣❡❧♦ ♠♦❞❡❧♦
é ❞❛❞♦ ♣♦r
et =xt−
"
a0+
q
X
k=1
akxt−k #
, ✭✷✳✸✮
♦♥❞❡ q é ❛ ♦r❞❡♠ ❞♦ ♠♦❞❡❧♦ ❡ ♦s ❝♦❡✜❝✐❡♥t❡ak sã♦ ♦❜t✐❞♦s ♣♦r ♠í♥✐♠♦s q✉❛❞r❛❞♦s✳
❯♠❛ ❛♠♦str❛ ❜❛st❛♥t❡ s✐♠♣❧❡s ❞♦ ❡❢❡✐t♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ✜❧tr♦ ❧✐♥❡❛r é ❞❛❞♦ ♣♦r
❚❤❡✐❧❡r ❡ ❊✉❜❛♥❦ ✭✶✾✾✸✮ ❡ r❡♣r♦❞✉③✐❞♦ ❛q✉✐✳ ❯t✐❧✐③❛♥❞♦ ♦ ❝♦♠♣♦♥❡♥t❡x(t) ❞❡ ✉♠ ♠❛♣❛
❞❡ ❍é♥♦♥ ❡ ♣❛ss❛♥❞♦ s❡✉s ✈❛❧♦r❡s ♣♦r ✉♠ ✜❧tr♦ ❞❡ ♦r❞❡♠ q = 6✱ ❡s❝♦❧❤✐❞♦ ❝♦♠ ❜❛s❡
♥♦ ❝r✐tér✐♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❆❦❛✐❦❡ ✭❆■❈✮✳ ❋✐❝❛ ❡✈✐❞❡♥t❡ ♣❡❧❛s ❋✐❣✉r❛s ✭✷✮ ❡ ✭✸✮ q✉❡ ♦ ❛tr❛t♦r é ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐st♦r❝✐❞♦ ♣❡❧❛ ❛♣❧✐❝❛çã♦ ❞♦ ✜❧tr♦✱ ❡s❝♦♥❞❡♥❞♦ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ s❡✉ ❞❡t❡r♠✐♥✐s♠♦✳
➱ ✐♠♣♦rt❛♥t❡ ❞✐s❝♦rr❡r r❛♣✐❞❛♠❡♥t❡ s♦❜r❡ ♦s ❡❢❡✐t♦s ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✜❧tr♦s ❧✐♥❡❛r❡s ❡s♣❡❝✐✜❝❛♠❡♥t❡ s♦❜r❡ ❛ ❡st❛tíst✐❝❛ ❇❉❙✳ ❆ ✜❧tr❛❣❡♠ ❢♦r♥❡❝❡ ✉♠❛ s♦❧✉çã♦ s✐♠♣❧❡s ♣❛r❛ s❡ t❡st❛r ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞❡ ✉♠❛ sér✐❡ t❡♠♣♦r❛❧✳ ❖r❛✱ ❝♦♠♦ ♦s r❡sí❞✉♦s ❞❡ ✉♠❛ sér✐❡ ✜❧tr❛❞❛ ♥ã♦ t❡rã♦ ❝♦rr❡❧❛çõ❡s ❧✐♥❡❛r❡s✱ ❡♥tã♦ q✉❛✐sq✉❡r ❝♦rr❡❧❛çã♦ ❡♥❝♦♥tr❛❞❛s ♥❡❧❡s ❞❡✲ ✈❡rã♦ t❡r ♦r✐❣❡♠ ♥ã♦✲❧✐♥❡❛r✳ ❉✐❛♥t❡ ❞✐ss♦✱ ❛ ❡st❛tíst✐❝❛ ❇❉❙ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ t❡st❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞♦s r❡sí❞✉♦s✳ ▼❛s✱ ❚❤❡✐❧❡r ❡ ❊✉❜❛♥❦ ✭✶✾✾✸✮ ♠♦str❛ q✉❡ t❡st❡s ❝✉❥♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ❝♦♥s✐st❡ ♥❡ss❛ ✜❧tr❛❣❡♠ ❛♣r❡s❡♥t❛rã♦ ✉♠ ❞❡s❡♠♣❡♥❤♦ ❞❡❣r❛❞❛❞♦ q✉❛♥❞♦ ♦s ❞❛❞♦s sã♦ ❝❛ót✐❝♦s✱ s✉❣❡r✐♥❞♦ s✉❛ ❛♣❧✐❝❛çã♦ s♦❜r❡ ♦s ❞❛❞♦s ♦r✐❣✐♥❛✐s ❡ ❝♦♠♣❛r❛❞❛ ❝♦♥tr❛ ❝♦♥❥✉♥t♦s ❞❡ s✉rr♦❣❛t❡s ❛♣r♦♣r✐❛❞♦s✳
✷✳✶ ❉❛❞♦s ✷✶
❋✐❣✉r❛ ✷✿ ❈♦♠♣♦♥❡♥t❡ ①✭t✮ ❞♦ ❛tr❛t♦r ❞❡ ❍é♥♦♥✳
✷✳✶ ❉❛❞♦s ✷✷
❞♦ ✜❧tr♦✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ t❡st❛r ❞✐✈❡rs♦s ✈❛❧♦r❡s ♣❛r❛ q ✈❡r✐✜❝❛♥❞♦ s❡✉ ❡❢❡✐t♦ s♦❜r❡ ❛s
✷✸
✸ ❘❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛r ❛s ♠❡t♦❞♦❧♦❣✐❛s ✉t✐❧✐③❛❞❛s ♥❛ r❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ♦♥❞❡ ❛ ❞✐♥â♠✐❝❛ ❡st✉❞❛❞❛ s❡ ❞❡s❡♥✈♦❧✈❡✳ ■♥✐❝✐❛♠♦s ❛ ❞✐s❝✉ssã♦ ❝♦♠ ❛s♣❡❝t♦s ❣❡r❛✐s ❡♥✈♦❧✈✐❞♦s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞♦s ♣❛ss♦s s❡❣✉✐❞♦s ❡ ❞♦s r❡s✉❧t❛❞♦s ❡s♣❡r❛❞♦s ❡ ♦❜t✐❞♦s ♥❡ss❛ ❡t❛♣❛✳
❖ ♣r✐♠❡✐r♦ ♣❛ss♦ ❛♦ ❜✉s❝❛r♠♦s ❡✈✐❞ê♥❝✐❛s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡t❡r♠✐♥íst✐❝♦ ❡♠ sér✐❡s t❡♠♣♦r❛✐s ✜♥❛♥❝❡✐r❛s é r❡❝♦♥str✉✐r ❛ ❞✐♥â♠✐❝❛ ❞♦ s✐st❡♠❛ q✉❡ ❛ ♦r✐❣✐♥♦✉✳ P♦ré♠✱ ❝♦♠♦ ❝❛♣t✉r❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ t♦❞♦ ✉♠ s✐st❡♠❛ t❡♥❞♦ à ♥♦ss❛ ❞✐s♣♦s✐çã♦ ❛♣❡♥❛s ✉♠ ✈❡t♦r ❝♦♠♣♦st♦ ♣♦r q✉❛♥t✐❞❛❞❡s ❡s❝❛❧❛r❡s❄ ❖ t❡♦r❡♠❛ q✉❡ ♥♦s ❣❛r❛♥t❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ t❛❧ ❛♥á❧✐s❡✱ ♦✉ s❡❥❛✱ q✉❡ ♥♦s ♣❡r♠✐t❡ ❛ tr❛♥s✐çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦❜s❡r✈❛çõ❡s ❡s❝❛❧❛r❡s ♣❛r❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ♠✉❧t✐✈❛r✐❛❞♦ é ❛tr✐❜✉í❞♦ ❛ ❚❛❦❡♥s ✭✶✾✽✶✮✱ s❡♥❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦
t❡♦r❡♠❛ ❞❡ ✐♠❡rsã♦✳ ❙✉♣♦♥❤❛ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ x(n) → (x(n)) = x(n + 1)✱ ♦♥❞❡
♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ x(t) é ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧✳ ❙❡ ❢♦r♠♦s ❝❛♣❛③❡s ❞❡ ♦❜s❡r✈❛r ✉♠❛ ú♥✐❝❛
q✉❛♥t✐❞❛❞❡ ❡s❝❛❧❛rh(·) ❞❡ ❛❧❣✉♠❛ ❢✉♥çã♦ ✈❡t♦r ❞❛s ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s g(x(n))✱ ❡♥tã♦ ❛
❡str✉t✉r❛ ❣❡♦♠étr✐❝❛ ❞❛ ❞✐♥â♠✐❝❛ ♠✉❧t✐✈❛r✐❛❞❛ ♣♦❞❡ s❡r ❞❡s❞♦❜r❛❞❛ ❛ ♣❛rt✐r ❞❡ss❡ ❝♦♥✲
❥✉♥t♦h(g(x(n)))❞❡ ♠❡❞✐❞❛s ❡s❝❧❛r❡s ❡♠ ✉♠ ♥♦✈♦ ❡s♣❛ç♦✱ ❝r✐❛❞♦ ❝♦♠ ♥♦✈♦s ✈❡t♦r❡s✱ ❝✉❥♦s
❝♦♠♣♦♥❡♥t❡s ❝♦♥s✐st❡♠ ❞❡ h(·)❛♣❧✐❝❛❞♦ ❛ ♣♦tê♥❝✐❛s ❞❡ g(x(n))✳ ❆ss✐♠✱ ❡ss❡s ✈❡t♦r❡s
y(n) = [h(x(n)), h(gT1
(x(n))), h(gT2
(x(n))),· · ·, h(gTd−1
(x(n)))], ✭✸✳✶✮
❞❡♥tr♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ ❝♦♠ s✉✜❝✐❡♥t❡s d ❞✐♠❡♥sõ❡s✱ r❡❝r✐❛rã♦ ❛ ♠❛✐♦r✐❛ ❞❛s
♣r♦♣r✐❡❞❛❞❡s ❞♦ s✐♥❛❧ ♠✉❧t✐✈❛r✐❛❞♦ ❞❡s❝♦♥❤❡❝✐❞♦x(n)♥♦ ♥♦✈♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧y(n)✱ ♣❛rt✐❝✲
✉❧❛r♠❡♥t❡✱ ❚❛❦❡♥s ✭✶✾✽✶✮ ♥♦s ❣❛r❛♥t❡ q✉❡ ❛ s❡qüê♥❝✐❛ ❞♦s ♣♦♥t♦sy(n)→y(n+1)✱ ♦✉ s❡❥❛✱
❛ ❡✈♦❧✉çã♦ ❞♦ s✐st❡♠❛ ♥♦ t❡♠♣♦✱ é ❛ ♠❡s♠❛ ❞❛ ❞✐♥â♠✐❝❛ ❞❡s❝♦♥❤❡❝✐❞❛x(n)→x(n+ 1)✳
✸ ❘❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ✷✹
❈♦♠ ✐ss♦✱ ❢❛③❡♥❞♦ ❛ ❢✉♥çã♦ ❡s❝❛❧❛r h(·)❝♦♠♦ s❡♥❞♦ ❛ ✈❛r✐á✈❡❧ ❡s❝❛❧❛r ♦❜s❡r✈❛❞❛ s(n)
h(x(n)) =s(n), ✭✸✳✷✮
❡ ❡s❝♦❧❤❡♥❞♦ ♣❛r❛ ❛ ❢✉♥çã♦ ❣❡r❛❧ g(x) ❛ ♦♣❡r❛çã♦ q✉❡ ❧❡✈❛ ❛❧❣✉♠ ✈❡t♦r ✐♥✐❝✐❛❧ x ♣❛r❛
❛❧❣✉♠ t❡♠♣♦ ❞❡❢❛s❛❞♦ τs ♣♦st❡r✐♦r✱ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ❛ Tk✲❡♥és✐♠❛ ♣♦tê♥❝✐❛ ❞❡ g(x)
t♦r♥❛✲s❡
gTk(x(n)) =x(n+T
k) =x(t0+ (n+Tk)τs), ✭✸✳✸✮
❡♥tã♦ ♦s ❝♦♠♣♦♥❡♥t❡s ❞❡y(n)❛ss✉♠❡♠ ❛ ❢♦r♠❛
y(n) = [s(n), s(n+T1), s(n+T2), . . . , s(n+Td−1)]. ✭✸✳✹✮
❆❜❛r❜❛♥❡❧ ✭✶✾✾✺✮ s✐♠♣❧✐✜❝❛ ❛✐♥❞❛ ♠❛✐s ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ t❡♦r❡♠❛✱ s✉❣❡r✐♥❞♦ ❝♦♠♦
❡s❝♦❧❤❛ ♣❛r❛ Tk = kT✱ ♦✉ s❡❥❛✱ q✉❡ ❛s ❞❡❢❛s❛❣❡♥s s❡❥❛♠ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♠ú❧t✐♣❧♦ ❞❡ ✉♠
✈❛❧♦rT✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ♦s ✈❡t♦r❡sy(n) t♦r♥❡♠✲s❡
y(n) = [s(n), s(n+T), s(n+ 2T), . . . , s(n+T(d−1))]. ✭✸✳✺✮
❊ss❡ ♥♦✈♦ ❡s♣❛ç♦ é r❡❧❛❝✐♦♥❛❞♦ ❛♦ ❡s♣❛ç♦ ♦r✐❣✐♥❛❧ ❞❡ x(n) ❛tr❛✈és ❞❡ tr❛♥s❢♦r♠❛çõ❡s
❝♦♥tí♥✉❛s ❡ ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❉❡ ❢❛t♦✱ ♦ ♦❜❥❡t♦ ❣❡♦♠étr✐❝♦ ✭♦ ❛tr❛t♦r✱ ✉t✐❧✐③❛♥❞♦ ♦s t❡r♠♦s
❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✮ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡sf(n) é ♣r❛t✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❛tr❛t♦r
❞♦ ❡s♣❛ç♦ ♦r✐❣✐♥❛❧ s❡✱ ❝♦♠♦ ❥á ♠❡♥❝✐♦♥❛❞♦✱ d ❢♦r ❣r❛♥❞❡ ♦ ❜❛st❛♥t❡✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱
♦ t❡♦r❡♠❛ ❞❡ ✐♠❡rsã♦ ♠❡♥❝✐♦♥❛ q✉❡✱ ❝❛s♦ ❛ ❞✐♥â♠✐❝❛ s❡ ❡♥❝♦♥tr❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡dA
❞✐♠❡♥sõ❡s ✭♦♥❞❡dA♣♦❞❡ s❡r ❢r❛❝✐♦♥❛❧✮✱ ❡♥tã♦ ❡s❝♦❧❤❡r ❛ ❞✐♠❡♥sã♦ ✐♥t❡✐r❛d❞♦ ❡s♣❛ç♦ ❞❡s✲
❞♦❜r❛❞♦✱ ♦♥❞❡ d >2dA✱ s❡rá s✉✜❝❡♥t❡ ♣❛r❛ r❡♠♦✈❡r q✉❛✐sq✉❡r ❛♠❜✐❣ü✐❞❛❞❡s r❡s✉❧t❛♥t❡s
❞❛ ♣r♦❥❡çã♦ ❞❛s ♦❜s❡r✈❛çõ❡s s♦❜r❡h(·)♦✉s(n)✳ ❆✐♥❞❛✱ ♣♦❞❡ s❡r ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ✉♠❛
❞✐♠❡♥sã♦ ♥❡❝❡ssár✐❛✱ ♠❡♥♦r q✉❡ ❛ s✉✜❝✐❡♥t❡d >2dA✳
❚❛❧ ❞✐s❝✉ssã♦ ❡♠ t♦r♥♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❞✐♠❡♥sõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ s❡ ❞❡s❞♦❜r❛r ♦ ❛tr❛t♦r é ❢✉♥❞❛♠❡♥t❛❧ à ❛♣❧✐❝❛çã♦ ♣rát✐❝❛ ❞♦ ❢❡rr❛♠❡♥t❛❧ ❛q✉✐ ❛♣r❡s❡♥t❛❞♦ ❛ sér✐❡s t❡♠♣♦r❛✐s ✜♥❛♥❝❡✐r❛s ❞❡✈✐❞♦ à ❡①✐stê♥❝✐❛ ❞❡ ✧r✉í❞♦s✧♥❛ sér✐❡✳ ❉❡✜♥✐♠♦s r✉í❞♦ ❝♦♠♦ s❡qüê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ❡ q✉❡ ♣♦ss✉❡♠ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ❞✐♠❡♥sõ❡s✱ ♦✉ s❡❥❛✱ ✉♠❛ sér✐❡ ❝♦♥t❡♥❞♦ r✉í❞♦s s❡♠♣r❡ ♣♦❞❡rá s❡r ❞❡s❞♦❜r❛❞❛ ♥✉♠❛ ❞✐♠❡♥sã♦ s✉♣❡r✐♦r àq✉❡❧❛ ♥❡❝❡ssár✐❛ à ♦❜s❡r✈❛çã♦ ❞❛ ❞✐♥â♠✐❝❛ ♥❡❧❛ ❝♦♥t✐❞❛✳ ❈♦♠ ✐ss♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ té❝♥✐❝❛ ✉s❛❞❛✱ ❛ ❡s❝♦❧❤❛
❞❡ ✉♠ ♥ú♠❡r♦ ❡①❝❡ss✐✈❛♠❡♥t❡ ❣r❛♥❞❡ ❞❡d❞✐♠❡♥sõ❡s ❛❝r❡s❝❡♥t❛ r❡❞✉♥❞â♥❝✐❛ ✭♥♦ ❝❛s♦ ❞❡
✸✳✶ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❞❡❢❛s❛❣❡♠ ✷✺
✸✳✶ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❞❡❢❛s❛❣❡♠
❆♣❡s❛r ❞♦ t❡♦r❡♠❛ ❞❡ ✐♠❡rsã♦ ❚❛❦❡♥s ✭✶✾✽✶✮ ♥♦s ❢♦r♥❡❝❡r ♦ ❢❡rr❛♠❡♥t❛❧ ❜ás✐❝♦ ♣❛r❛ ❛ r❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ❡❧❡ ♥ã♦ ♠❡♥❝✐♦♥❛❞❛ ♥❛❞❛ ❛ r❡s♣❡✐t♦ ❞❛ ❡s❝♦❧❤❛ ❞❛
❞❡❢❛s❛❣❡♠ ❞❡ t❡♠♣♦ T ❛ s❡r ✉s❛♥❞♦ ♥♦ ♣r♦❝❡ss♦✳ ❉✐❛♥t❡ ❞❡ss❛ ♦♠✐ssã♦ ❡ ❞❛ ✐♥❡①✐stê♥❝✐❛
❞❡ q✉❛❧q✉❡r t❡♦r❡♠❛ q✉❡ ♥♦s ❢♦r♥❡ç❛ ✉♠ ❡♠❜❛s❛♠❡♥t♦ ♣❛r❛ ❡ss❛ ❡s❝♦❧❤❛✱ ❆❜❛r❜❛♥❡❧ ✭✶✾✾✺✮ ❞✐s❝✉t❡ ❡①t❡♥s✐✈❛♠❡♥t❡ ♦ ♣r♦❜❧❡♠❛✱ ♣❡r♠✐t✐♥❞♦ ❛ ❡❧❛❜♦r❛❝ã♦ ❞❡ ✉♠❛ ✧❧✐st❛✧❞❡ r❡q✉✐s✐t♦s
q✉❡T ❞❡✈❡ ❛t❡♥❞❡r✿
✐ ❊❧❡ ❞❡✈❡ s❡r ❛❧❣✉♠ ♠ú❧t✐♣❧♦ ❞♦ t❡♠♣♦ ❞❡ ❛♠♦str❛❣❡♠ τs✱ ✉♠❛ ✈❡③ q✉❡ t❡♠♦s ❞❛❞♦s
❛♣❡♥❛s ♥❡ss❡s t❡♠♣♦s✳
✐✐ ❙❡ ❛ ❞❡❢❛s❛❣❡♠ ❢♦r ♠✉✐t♦ ❝✉rt❛✱ ❛s ❝♦♦r❞❡♥❛❞❛ss(n)❡s(n+T)q✉❡ ❞❡s❡❥❛♠♦s ✉s❛r ♥♦
✈❡t♦r r❡❝♦♥str✉í❞♦ yn ♥ã♦ s❡rã♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦✉ s❡❥❛✱ ♥ã♦ ❤❛✈❡rá
♣❛ss❛❞♦ t❡♠♣♦ s✉✜❝✐❡♥t❡ ❡♥tr❡ ❛s ♦❜s❡r✈❛çõ❡s ♣❛r❛ q✉❡ ♦ s✐st❡♠❛ t❡♥❤❛ ❡✈♦❧✉í❞♦ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ♥♦✈❛s ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞❛ s✉❛ ❞✐♥â♠✐❝❛ ♣♦ss❛♠ s❡r ♦❜t✐❞❛s✳
✐✐✐ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ s❡T ❢♦r ❡①❝❡ss✐✈❛♠❡♥t❡ ❧♦♥❣♦✱s(n)❡ s(n+T)♣❛r❡❝❡rã♦ ❝♦♠♣❧❡t❛✲
♠❡♥t❡ ❛❧❡❛tór✐❛s ❡♠ r❡❧❛çã♦ ✉♠❛ à ♦✉tr❛ ❞❡✈✐❞♦ à ✐♥st❛❜✐❧✐❞❛❞❡ ✐♥trí♥s❡❝❛ ❞♦s s✐st❡♠❛s ❝❛ót✐❝♦s✳
❈♦♠ ❡ss❛ ❧✐st❛ ❡♠ ♠ã♦s✱ ♣❡r❝❡❜❡✲s❡ q✉❡ ❛ ♠❡t♦❞♦❧♦❣✐❛ ♣❛r❛ s❡ ❝❤❡❣❛r à ♠❡❧❤♦r ❡s❝♦❧❤❛
♣❛r❛T ♣♦❞❡ s❡r ❜❛s❡❛❞❛ ❡♠ ✉♠ ❛s♣❡❝t♦ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ♣ró♣r✐♦ ❝❛♦s✿ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡
❣❡r❛r ✐♥❢♦r♠❛çã♦✳
❙✉♣♦♥❤❛ q✉❡ t❡♥❤❛♠♦s ✉♠❛ r❡s♦❧✉çã♦ ✜①❛ ❡ ✜♥✐t❛ ❡♠ ♥♦ss♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ s❡♥❞♦
✐♠♣♦ssí✈❡❧ ❞✐st✐♥❣✉✐r ❞♦✐s ♣♦♥t♦sx1(t)❡x2(t)✱ ❧♦❝❛❧✐③❛❞♦s ❞❡♥tr♦ ❞❡ ✉♠❛ ❜♦❧❛ ❞❡ t❛♠❛♥❤♦
R♥♦ ♠♦♠❡♥t♦t✳ ❊♠ ❛❧❣✉♠ t❡♠♣♦ ♣♦st❡r✐♦rt′ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡ss❡s ♣♦♥t♦s ❝r❡s❝❡✉ ♣❛r❛
|x1(t′
)−x2(t′
)| ≈ |x1(t)−x2(t)|exp[λ|t′
−t|];λ >0✳ ❙❡ ❡ss❛ ❞✐stâ♥❝✐❛ ❢♦r ♠❛✐♦r q✉❡ R✱
❡♥tã♦ s❡r❡♠♦s ❝❛♣❛③❡s ❞❡ ❞✐st✐♥❣✉✐r ❡♥tr❡ ❛s ór❜✐t❛s x1(t′
) ❡ x2(t′
)✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❛
✐♥st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ r❡✈❡❧♦✉ ✉♠❛ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ q✉❡ ♥ã♦ ❡st❛✈❛ ❞✐s♣♦♥í✈❡❧ ❛♥t❡r✐♦r♠❡♥t❡ ❛♦ ♥♦s ♣❡r♠✐t✐r ♦❜s❡r✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞✉❛s ór❜✐t❛s ♦♥❞❡ ❛♥t❡s ❛❝r❡❞✐tá✈❛♠♦s ❤❛✈❡r ❛♣❡♥❛s ✉♠❛✳
❊ss❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❣❡r❛çã♦ ❞❡ ✐♥❢♦r♠❛çã♦ é ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❛♥á❧✐s❡ ❞❡ s✐st❡♠❛s
❞✐♥â♠✐❝♦s ♥ã♦✲❧✐♥❡❛r❡s ♣♦✐s✱ ❝❛s♦ ♦ s✐st❡♠❛ ❢♦ss❡ ❧✐♥❡❛r ❡ ❡stá✈❡❧ ❣❧♦❜❛❧♠❡♥t❡✱ t♦❞♦s ♦s λ
✸✳✶ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❞❡❢❛s❛❣❡♠ ✷✻
❊♠ ✉♠ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r ❝♦♠d ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ ❤❛✈❡rã♦ d❡①♣♦❡♥t❡s ❝♦♠♦ λ1 >
λ2, . . . , > λd✱ s❡♥❞♦ q✉❡ ❛ s✉❛ s♦♠❛ s❡rá ♥❡❣❛t✐✈❛ ❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s s❡rá ♣♦s✐t✐✈♦✳ P♦r ✉♠ ❧❛❞♦✱ ❡ss❡ ❡①♣♦❡♥t❡ ♣♦s✐t✐✈♦ ♣r♦♠♦✈❡ ♦ ❛❢❛st❛♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛s tr❛❥❡tór✐❛s✳ P♦ré♠✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ❛ s♦♠❛ ❞♦s ❡①♣♦❡♥t❡s é ♥❡❣❛t✐✈❛✱ ♦✉ s❡❥❛✱ ❛s tr❛❥❡tór✐❛s ♥♦ ❛tr❛t♦r ♣❡r♠❛♥❡❝❡rã♦ ❝♦♥✜♥❛❞❛s ❛ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❊ss❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛❢❛st❛♠❡♥t♦ ❡ ❝♦♥✜♥❛♠❡♥t♦ ❞❛ tr❛❥❡tór✐❛ é q✉❡ ♦r✐❣✐♥❛ ❛ ❞✐♥â♠✐❝❛ ❝❛ót✐❝❛ ❞❡ ✉♠ s✐st❡♠❛✳
✸✳✶✳✶ ❊①♣❛♥sã♦ ❡ ❈♦♥tr❛çã♦ ❞♦ ❆tr❛t♦r
■♥t✉✐t✐✈❛♠❡♥t❡✱ ❛ r❡s♣♦st❛ ❛ ❡ss❛ ❛♣❛r❡♥t❡ ❝♦♥tr❛❞✐çã♦ ✕ ❝♦♠♦ tr❛❥❡tór✐❛s ♣♦❞❡♠ s❡ ❛❢❛st❛r ❡ ❛✐♥❞❛ ❛ss✐♠ ♣❡r♠❛♥❡❝❡r❡♠ ❞❡♥tr♦ ❞❡ ✉♠❛ r❡❣✐ã♦ ❝♦♠♣❛❝t❛❄ ✕ ❢♦✐ ❞❛❞❛ ♣♦r ❍é♥♦♥ ✭✶✾✼✻✮ ❡ ❘öss❧❡r ✭✶✾✼✻✮✱ ❛tr❛✈és ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❡①♣❛♥sã♦ ❡ ❝♦♥tr❛❝ã♦✳
P♦r ❡①❡♠♣❧♦✱ s❡ t♦♠❛r♠♦s ✉♠ r❡tâ♥❣✉❧♦abcd❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱
✉♠ ❛tr❛t♦r ❡str❛♥❤♦ s✉r❣✐rá q✉❛♥❞♦ ❛ tr❛❥❡tór✐❛ ❝♦♥tr❛✐r ♦ r❡tâ♥❣✉❧♦ ❡♠ ❛❧❣✉♠❛s ❞✐r❡çõ❡s ✭♦✉ s❡❥❛✱ ♦ ❡❢❡✐t♦ ❞❛ ❞✐ss✐♣❛çã♦✮ ❡ ❡st✐❝á✲❧♦ ❡♠ ♦✉tr❛s ✭♦ ❛❢❛st❛♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✮✳ P♦ré♠✱ ❡ss❛ ❡①♣❛♥sã♦ ♥ã♦ ♣♦❞❡ ❝♦♥t✐♥✉❛r ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ♣♦✐s✱ ♣❛r❛ q✉❡ ♦ r❡tâ♥❣✉❧♦ ♣❡r♠❛♥❡ç❛ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ❝♦♠♣❛❝t❛ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ❡❧❡ ❞❡✈❡ s❡r ❞♦❜r❛❞♦ s♦❜r❡ s✐ ♠❡s♠♦✳
❆ss✐♠✱ ♦ r❡tâ♥❣✉❧♦abcdé ❛❝❤❛t❛❞♦✱ ❡①♣❛♥❞✐❞♦ ❡ ❝♦♥tr❛í❞♦ ♥✉♠❛ ❢❡rr❛❞✉r❛✶a′
, b′
, c′
, d′✱
❝♦♠♦ ♠♦str❛❞♦ ❡♠ S1✳ ❙❡ r❡♣❡t✐r♠♦s ♦ ♣r♦❝❡ss♦✱ S1 s❡ tr❛♥s❢♦r♠❛rá ❡♠ S2 ❡ ❛ss✐♠
s✉❝❡ss✐✈❛♠❡♥t❡✳ ❆ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ S∞ ❝♦♥s✐st✐rá ❡♠ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ❝❛♠❛❞❛s
s❡♣❛r❛❞❛s ♣♦r ❞✐stâ♥❝✐❛s ❞❡ ❞✐✈❡rs♦s t❛♠❛♥❤♦s q✉❡✱ s❡ ❢♦r ❢❡✐t❛ ✉♠❛ s❡❝çã♦ tr❛♥s✈❡rs❛❧ ❞♦ ♠❡s♠♦✱ ❝❤❡❣❛r❡♠♦s ❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ❡str✉t✉r❛ ❢r❛❝t❛❧ ❞♦ ❛tr❛t♦r✱ ❝♦♥s❡qüê♥❝✐❛
❞❛s ❡①♣❛♥sõ❡s ❡ ❝♦♥tr❛çõ❡s q✉❡ ❣❡r❛r❛♠S∞✳
✸✳✶✳✷ ■♥❢♦r♠❛çã♦ ♠út✉❛ ♠é❞✐❛
❖ ❝♦♥❝❡✐t♦ ❞❡ ✐♥❢♦r♠❛çã♦ ♠út✉❛ ✕ ✐♥tr♦❞✉③✐♥❞♦ ♣♦r ❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ✕ ❡♥tr❡ ❞✉❛s
♠❡❞✐❞❛sai ❡bj s♦rt❡❛❞❛s ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ♣♦ssí✈❡✐s ♠❡❞✐❞❛sA❡ B r❡❢❡r❡✲s❡ à q✉❛♥t✐❞❛❞❡
❞❡ ✐♥❢♦r♠❛çã♦ q✉❡ ♣♦❞❡ s❡r ❛♣r❡♥❞✐❞❛ ❛ ♣❛rt✐r ❞❡ ai s♦❜r❡ ❛ ♠❡❞✐❞❛ ❞❡ bj✳ ❊♠ t❡r♠♦s
❜✐♥ár✐♦s✱ ✐ss♦ s❡r✐❛
log2
PAB(ai, bj)
PA(ai)PB(bj)
, ✭✸✳✻✮
♦♥❞❡PAB(a, b)é ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❥✉♥t❛ ♣❛r❛ ♠❡❞✐❞❛sA❡B q✉❡ r❡s✉❧t❛♠
♥♦s ✈❛❧♦r❡s a ❡ b✳ PA(a) ❡ PB(b) sã♦ ❛s ❞❡♥s✐❞❛❞❡s ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ✐♥❞✐✈✐❞✉❛✐s ♣❛r❛
✸✳✶ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❞❡❢❛s❛❣❡♠ ✷✼
❛s ♠❡❞✐❞❛s ❞❡A ❡ B✳ ❆ss✐♠✱ s❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ✈❛❧♦r ❞❡ A q✉❡ r❡s✉❧t❛ ❡♠ ai ❢♦r ❝♦♠✲
♣❧❡t❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♠❡❞✐❞❛ ❞❡B q✉❡ r❡s✉❧t♦✉ ❡♠ bj✱ ❡♥tã♦ PAB(a, b) s❡ t♦r♥❛rá
PA(a)PB(b)❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çã♦ ❡♥tr❡ ❛s ♠❡❞✐❞❛s✱ ♦✉ s❡❥❛✱ ❛ ✐♥❢♦r♠❛çã♦ ♠út✉❛✱
s❡rá ③❡r♦✳ ❆ ♠é❞✐❛ ❞❡ t♦❞❛s ❛s ♠❡❞✐❞❛s ❞❡ss❛ ❡st❛tíst✐❝❛ ✐♥❢♦r♠❛❝✐♦♥❛❧ é ❛ ✐♥❢♦r♠❛çã♦
♠út✉❛ ♠é❞✐❛ ❡♥tr❡ ❛s ♠❡❞✐❞❛sA ❡ B ❡ é ❞❛❞❛ ♣♦r
IAB =
X
ai,bj
PAB(ai, bj) log2
PAB(ai, bj)
PA(ai)PB(bj)
. ✭✸✳✼✮
❈♦♠ ❡ss❛ q✉❛♥t✐❞❛❞❡✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠ ❝r✐tér✐♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛
♠út✉❛ ❡♥tr❡ ♠❡❞✐❞❛ss(t)t✐r❛❞❛s ❡♠ ✉♠ t❡♠♣♦t ❡ ♠❡❞✐❞❛ss(t+T)❡♠ ✉♠ t❡♠♣♦t+T✳
❙❡ ❛ss✉♠✐r♠♦s ❛s ♦❜s❡r✈❛çõ❡ss(n) ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s A ❡ ♦s ✈❛❧♦r❡s ❞❡
s(n+T) ❝♦♠♦ s❡♥❞♦ ❛s ♠❡❞✐❞❛s B✱ ❡♥tã♦ ❛ ✐♥❢♦r♠❛çã♦ ♠út✉❛ ♠é❞✐❛ ❡♥tr❡ ❡ss❡s ❞♦✐s
❝♦♥❥✉♥t♦s s❡rá
I(T) = X
s(n),s(n+T)
P(s(n), s(n+T)) log2
P(s(n), s(n+T))
P(s(n))P(s(n+T))
. ✭✸✳✽✮
❉❡ ❛❝♦r❞♦ ❝♦♠ ❋r❛s❡r ❡ ❙✇✐♥♥❡② ✭✶✾✽✻✮✱ ❛ ❢✉♥çã♦I(T)♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♠♦ ✉♠❛ ❡s♣é❝✐❡
❞❡ ❢✉♥çã♦ ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♥ã♦✲❧✐♥❡❛r ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s q✉❛♥❞♦ ♦s ✈❛❧♦r❡s ❞❡ s(n) ❡
s(n+T)sã♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tr❡ s✐ ♣❛r❛ s❡r❡♠ út❡✐s ❝♦♠♦ ❝♦♦r❞❡♥❛❞❛s ❡♠
✉♠ ✈❡t♦r ❞❡❢❛s❛❞♦ ♥♦ t❡♠♣♦✱ ♠❛s ♥ã♦ tã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❛ ♣♦♥t♦ ❞❡ ♥ã♦ t❡r❡♠ ♥❡♥❤✉♠ t✐♣♦ ❞❡ r❡❧❛çã♦ ❡♥tr❡ s✐✳ ❆ s✉❣❡stã♦ ❞❛❞❛ ♣♦r ❋r❛s❡r ❡ ❙✇✐♥♥❡② ✭✶✾✽✻✮✱ ❆❜❛r❜❛♥❡❧ ✭✶✾✾✺✮ é
❛ ❞❡ t♦♠❛r♠♦s ❝♦♠♦T ♦ ♣r✐♠❡✐r♦ ♠í♥✐♠♦ ❞❡ I(T) ❡ ✉t✐❧✐③á✲❧♦ ♥❛ r❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦
❞❡ ❢❛s❡✳
❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❛ ✐♥❢♦r♠❛çã♦ ♠út✉❛ ♠é❞✐❛✱ ✉t✐❧✐③❛r❡♠♦s ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ▲❖❘❊◆❩ ✭✶✾✻✸✮ ❡ q✉❡ s❡r✈✐✉ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♥❛ ✈❡r✐✜❝❛çã♦ ❞❛ ✈❛❧✐❞❛❞❡ ❞♦s ❛❧❣♦r✐t♠♦s ❡s❝r✐t♦s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦ tr❛❜❛❧❤♦✱ ❜❛s❡❛❞♦s ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❡♠ ❆❜❛r❜❛♥❡❧ ✭✶✾✾✺✮✳ ❆s ❡q✉❛çõ❡s q✉❡ ❝♦♠♣õ❡♠ ♦ s✐st❡♠❛ sã♦
˙
x = σ(y−x)
˙
y = −xz+rx−y ✭✸✳✾✮
˙
z = xy−bz.
P❛r❛ ❛ ❣❡r❛çã♦ ❞♦s ❞❛❞♦s✱ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ♣❛râ♠❡tr♦s ♦s ✈❛❧♦r❡s σ = 16✱ b = 4 ❡
r= 45,92❡ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❘✉♥❣❡✲❑✉tt❛ ❞❡ q✉❛rt❛ ♦r❞❡♠ ❝♦♠ ✐♥t❡r✈❛❧♦s ❞❡ τs= 0,01✳
