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❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥

❋✐❧❡ ✸ ❏♦sé ❘♦❜❡rt♦ ❆♠❛③♦♥❛s

❥r❛❅❧❝s✳♣♦❧✐✳✉s♣✳❜r ❚❡❧❡❝♦♠♠✉♥✐❝❛t✐♦♥s ❛♥❞ ❈♦♥tr♦❧ ❊♥❣✐♥❡❡r✐♥❣ ❉❡♣t✳ ✲ P❚❈ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦ ✲ ❯❙P

❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶ ✴ ✻✼

❖✉t❧✐♥❡

✶

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ▼♦❞❡❧♦ ▼❲▼

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥

❖✉t❧✐♥❡

✶

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ▼♦❞❡❧♦ ▼❲▼

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥

❈❧❛ss❡s ♦❢ ♠♦❞❡❧✐♥❣

❚r❛✣❝ ♠♦❞❡❧s ♠❛② ❜❡ ❝❧❛ss✐✜❡❞ ❛s ❤❡t❡r♦❣❡♥♦✉s ♦r ❤♦♠♦❣❡♥❡♦✉s✳

❍❡t❡r♦❣❡♥❡♦✉s ♠♦❞❡❧s s✐♠✉❧❛t❡ t❤❡ ❛❣❣r❡❣❛t❡ tr❛✣❝ ✭tr❛✣❝ ❣❡♥❡r❛t❡❞ ❜② s❡✈❡r❛❧ ✉s❡rs✱ ♣r♦t♦❝♦❧s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✮ ♦✈❡r ❛ ♥❡t✇♦r❦ ❧✐♥❦✳ ❍♦♠♦❣❡♥❡♦✉s ♠♦❞❡❧s r❡❢❡r t♦ ❛ s♣❡❝✐✜❝ ❦✐♥❞ ♦❢ tr❛✣❝✱ ❛s t❤❡ ▼P❊● ✈✐❞❡♦ tr❛✣❝✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥

❈❧❛ss❡s ♦❢ ♠♦❞❡❧✐♥❣

❍❡t❡r♦❣❡♥❡♦✉s ♠♦❞❡❧s ♠❛② ❜❡ s✉❜❞✐✈✐❞❡❞ ✐♥ t✇♦ ❝❧❛ss❡s✿ ❜❡❤❛✈✐♦r❛❧ ♦r str✉❝t✉r❛❧✳

❇❡❤❛✈✐♦r❛❧ ♠♦❞❡❧s ♠♦❞❡❧ t❤❡ tr❛✣❝ st❛t✐st✐❝s✱ ❛s ❝♦rr❡❧❛t✐♦♥✱ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦r ❡✈❡♥ ❤✐❣❤❡r ♦r❞❡r st❛t✐st✐❝s ✭t❤✐r❞ ❛♥❞ ❢♦✉rt❤ ♦r❞❡rs✱ ❢♦r ❡①❛♠♣❧❡✮ ✇✐t❤♦✉t t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣❤②s✐❝❛❧ ♠❡❝❤❛♥✐s♠ ♦❢ tr❛✣❝ ❣❡♥❡r❛t✐♦♥ ✭✐✳ ❡✳✱ ❜❡❤❛✈✐♦r❛❧ ♠♦❞❡❧s✬ ♣❛r❛♠❡t❡rs ❛r❡ ♥♦t ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥s ♥❡t✇♦r❦✬s ♣❛r❛♠❡t❡rs✮✳ ❙tr✉❝t✉r❛❧ ♠♦❞❡❧s ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ♣❛❝❦❡ts ❣❡♥❡r❛t✐♦♥ ♠❡❝❤❛♥✐s♠s ❛♥❞ t❤❡✐r ♣❛r❛♠❡t❡rs ♠❛② ❜❡ ♠❛♣♣❡❞ t♦ ♥❡t✇♦r❦✬s ♣❛r❛♠❡t❡rs✱ ❛s ♥✉♠❜❡r ♦❢ ✉s❡rs ❛♥❞ ❜❛♥❞✇✐❞t❤✳

❋●◆✱ ❆❘❋■▼❆ ❛♥❞ ▼❲▼ ❛r❡ ❜❡❤❛✈✐♦r❛❧ ♠♦❞❡❧s ♦❢ ❛❣❣r❡❣❛t❡ tr❛✣❝✳ ❖♥✴❖✛ ♣r♦❝❡ss❡s ♠❛② ❜❡ ✉s❡❞ ❛s tr❛✣❝ str✉❝t✉r❛❧ ♠♦❞❡❧s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❖✉t❧✐♥❡

✶

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ▼♦❞❡❧♦ ▼❲▼

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋●◆ ♣r♦❝❡ss

❚❤❡ ❋●◆ ♣r♦❝❡ss✱ ♣r♦♣♦s❡❞ ❜② ▼❛♥❞❡❧❜r♦t ❛♥❞ ❱❛♥ ◆❡ss ✐♥ ✶✾✻✽ ❢♦r ♠♦❞❡❧✐♥❣ ▲❘❉ ❤②❞r♦❧♦❣✐❝❛❧ s❡r✐❡s✱ ✐s t❤❡ ✜rst ✐♠♣♦rt❛♥t ❧♦♥❣ ♠❡♠♦r② ♠♦❞❡❧ t❤❛t ❛♣♣❡❛rs ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ■❢ {①t}t∈Z ✐s ❛ ❋●◆✱ t❤❡♥ ①t ✐s ❛ st❛t✐♦♥❛r② ♣r♦❝❡ss ✇✐t❤ ❛✉t♦❝♦✈❛r✐❛♥❝❡ ❣✐✈❡♥ ❜② ❈①(τ) = σ✷①

✷ [|τ + ✶|✷❍ − ✷|τ|✷❍ + |τ − ✶|✷❍]✱

τ = . . . , −✶, ✵, ✶, . . .✮✳ ❚❤❡ ❋●◆ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✜rst ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss ❦♥♦✇♥ ❛s ❋r❛❝t✐♦♥❛r② ❇r♦✇♥✐❛♥ ▼♦t✐♦♥✱ ❋❇▼ {❇❍(t) : ✵ ≤ t ≤ ∞} ✇✐t❤ ❍✉rst ♣❛r❛♠❡t❡r ✵ < ❍ < ✶✱ ✐✳ ❡✳✱ ①t = ∆❇❍(t) = ❇❍(t + ✶) − ❇❍(t), t = ✵, ✶, ✷, . . . . ✭✶✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✼ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋❇▼ r❡❛❧✐③❛t✐♦♥s

200 400 600 800 1000 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 t Realizações de processos FBM H=0,6 H=0,5 H=0,7 H=0,8 H=0,9

❋✐❣✉r❡✿ ❘❡❛❧✐③❛t✐♦♥s ♦❢ ❋❇▼ ♣r♦❝❡ss❡s ❢♦r s❡✈❡r❛❧ ❍✉rst ♣❛r❛♠❡t❡rs ✈❛❧✉❡s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✽ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋❇▼ Pr♦❝❡ss

❚❤❡ ❋❇▼ ❤❛s ❛ s♣❡❝✐❛❧ ♥❛♠❡ ✇❤❡♥ ❍ = ✶/✷✿ ✭❇r♦✇♥✐❛♥ ♠♦t✐♦♥✮ ❛♥❞ ✐t ✐s ❞❡s✐❣♥❛t❡❞ ❜② ❇✶/✷(t)✳ ■♥ t❤✐s ❝❛s❡✱ ①✶, ①✷, ... ❛r❡ ✐♥❞❡♣❡♥❞❡♥t

• ❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳

❲❡ ❝❛♥ ❝r❡❛t❡ ❛ ❞✐s❝r❡t❡ t✐♠❡ ❋❇▼ ✭❉❋❇▼✮✱ ❞❡♥♦t❡❞ ❜② ❇t✱ ❜② ♠❡❛♥s ♦❢ t❤❡ ❝✉♠✉❧❛t✐✈❡ s✉♠ ♦❢ t❤❡ ❋●◆ {①t} s❛♠♣❧❡s✿ ❇t ≡ ❇❍(t) =

t−✶

• ✉=✵

①✉, t = ✶, ✷, . . . . ✭✷✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✾ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋❇▼ Pr♦❝❡ss ✲ ❝♦♥t✳

❚❤❡ ❉❋❇▼ ❙❉❋ ✐s ❣✐✈❡♥ ❜② P❇t(❢ ) = σ✷

①❈❍ ∞

• ❥=−∞

✶ |❢ + ❥|✷❍+✶ , −✶ ✷ ≤ ❢ ≤ ✶ ✷, ✭✸✮ ✐♥ ✇❤✐❝❤ σ✷

① ✐s t❤❡ ♣♦✇❡r ♦❢ ❛ ③❡r♦ ♠❡❛♥ ❋●◆✱ ❈❍ = Γ(✷❍+✶) s✐♥ (π❍) ✷π✷❍+✶

❛♥❞ ✵ < ❍ < ✶✳ ❆❝❝♦r❞✐♥❣ t♦ ✭✸✮✱ t❤❡ ❉❋❇▼ ❙❉❋ ❤❛s ❛ |❢ |−α✱ ✵ < α < ✶✱ s✐♥❣✉❧❛r✐t②✱ ✐♥ t❤❡ ♦r✐❣✐♥✱ ❛s P❇t ∝ |❢ |✶−✷❍, ❢ → ✵ . ✭✹✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✵ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋●◆✬s ❙❉❋

❚❤❡ ❋●◆ ❛♥❞ t❤❡ ❉❋❇▼ ❛r❡ r❡❧❛t❡❞ ❜② t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❍(③) = ❳(③) ❇(③) = ✶ − ③−✶, ✭✺✮ ✐♥ ✇❤✐❝❤ ❳(③) ❡ ❇(③) ❞❡♥♦t❡ t❤❡ ③✲tr❛♥s❢♦r♠s ♦❢ ①t ❛♥❞ ❇t✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❢r❡q✉❡♥❝② r❡s♣♦♥s❡ ❛ss♦❝✐❛t❡❞ t♦ ✭✺✮ ✐s ❍(❢ ) = ❍(③)|③=❡❥✷π❢ = ✶ − ❡−❥✷π❢ . ✭✻✮ ❆s t❤❡ ✐♥♣✉t✴♦✉t♣✉t r❡❧❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❙❉❋s ✐s P①(❢ ) = |❍(❢ )|✷P❇t(❢ ) , ✭✼✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✶ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋●◆✬s ❙❉❋ ✲ ❝♦♥t✳

|❍(❢ )|✷ ✐s ❣✐✈❡♥ ❜②✱ |❍(❢ )|✷ = ●(❢ ) = ✹ s✐♥✷ (π❢ ) , ✭✽✮ t❤❡♥ t❤❡ ❋●◆✬s ❙❉❋ ✐s ❡q✉❛❧ t♦ P①(❢ ) = ✹ s✐♥✷ (π❢ )P❇t(❢ ) . ✭✾✮ ✭✸✮ ❡ ✭✾✮ s❤♦✇ t❤❛t t❤❡ ❋●◆✬s ❙❉❋ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ♦♥❧② t✇♦ ♣❛r❛♠❡t❡rs✿σ✷

① ❡ ❍ ✭r❡s♣♦♥s✐❜❧❡ ❢♦r t❤❡ s♣❡❝tr✉♠ s❤❛♣❡✮✳

❚❤❡ ❋●◆ ✐s ❝♦♠♣❧❡t❡❧② s♣❡❝✐✜❡❞ ❜② ✐ts ♠❡❛♥ ❛♥❞ ❜② ✐ts ❙❉❋✱ ❛s ✐t ✐s

• ❛✉ss✐❛♥✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣

❋●◆✬s ❙❉❋ ✲ ❝♦♥t✳

■t ♠❛② ❜❡ s❤♦✇♥ t❤❛t ✭✾✮ ♠❛② ❜❡ r❡✇r✐tt❡♥ ❛s✿ P①(❢ ) = ❆(❢ , ❍)[|✷π❢ |−✷❍−✶ + ❇(❢ , ❍)] , ✭✶✵✮ ✐♥ ✇❤✐❝❤ ❆(❢ , ❍) = ✷ s✐♥ (π❍)Γ(✷❍ + ✶)(✶ − ❝♦s (✷π❢ )) ❡ ❇(❢ , ❍) = ∞

❥=✶[(✷π❥ + ✷π❢ )−✷❍−✶ + (✷π❥ − ✷π❢ )−✷❍−✶]✳

❋♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ ❢ ✱ P①(❢ ) ∝ |❢ |✶−✷❍ ✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❖✉t❧✐♥❡

✶

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ▼♦❞❡❧♦ ▼❲▼

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❋❚ ❡ ❲❋❚

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ ❛ s✐❣♥❛❧ ①(t)✱ ✐❢ ❡①✐sts✱ ✐s ❞❡✜♥❡❞ ❛s ❳(ν) = ❚❋{①(t)} = ∞

−∞

①(t)❡−❥✷πνt❞t , ✭✶✶✮ ✐♥ ✇❤✐❝❤ ν ❞❡♥♦t❡s t❤❡ ❢r❡q✉❡♥❝② ✐♥ ❝②❝❧❡s✴s❡❝♦♥❞ ❬❍③❪✳

• ❛❜♦r ❤❛s s❤♦✇♥ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ r❡♣r❡s❡♥t t❤❡ ❧♦❝❛❧ s♣❡❝tr❛❧

❝♦♥t❡♥t ♦❢ ❛ s✐❣♥❛❧ ①(t) ❛r♦✉♥❞ ❛♥ ✐♥st❛♥t ♦❢ t✐♠❡ τ ❜② t❤❡ ✇✐♥❞♦✇❡❞ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❲❋❚✮ ❳❚(ν, τ) = ∞

−∞

①(t)❣❚(t − τ)❡−❥✷πνt❞t , ✭✶✷✮ ✐♥ ✇❤✐❝❤ ❣❚(t) ✐s ❛ ✇✐♥❞♦✇ ♦❢ ✜♥✐t❡ ❞✉r❛t✐♦♥ s✉♣♣♦rt ❚ ❛♥❞ ν ❞❡♥♦t❡s ❢r❡q✉❡♥❝②✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❋❚ ❛♥❞ ❲❋❚ ✲ ❝♦♥t✳

❚❤❡ ❲❋❚ ✐s ❛ ❜✐✲❞✐♠❡♥s✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ t✐♠❡✲❢r❡q✉❡♥❝② ♣❧❛♥❡ ♦r ❞♦♠❛✐♥ ❛s ✐t ❞❡♣❡♥❞s ♦♥ t❤❡ ν ❛♥❞ τ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ❲❋❚ ✇♦✉❧❞ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ ❛ ❦✐♥❞ ♦❢ ❝♦♥t✐♥✉♦✉s ✏s❤❡❡t ♠✉s✐❝✑ ❞❡s❝r✐♣t✐♦♥ ♦❢ ①(t)✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❍❡✐s❡♥❜❡r❣✬s ✉♥❝❡rt❛✐♥t② ♣r✐♥❝✐♣❧❡✱ ❛ s✐❣♥❛❧ ✇❤♦s❡ ❡♥❡r❣② ❝♦♥t❡♥t ✐s q✉✐t❡ ✇❡❧❧ ❧♦❝❛❧✐③❡❞ ✐♥ t✐♠❡ ❤❛s t❤✐s ❡♥❡r❣② q✉✐t❡ s♣r❡❛❞ ♦✉t ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✳ ❆s t❤❡ ✇✐♥❞♦✇ ♦❢ ✭✶✷✮ ❤❛s ❛ ✜①❡❞ s✐③❡ ❚✱ ✇❡ ♠❛② ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❲❋❚ ✐s ♥♦t ❣♦♦❞ t♦ ❛♥❛❧②③❡ ✭♦r ✐❞❡♥t✐❢②✮ ❜❡❤❛✈✐♦rs ♦❢ ①(t) ♦❝❝✉rr✐♥❣ ✐♥ t✐♠❡ ✐♥t❡r✈❛❧s ♠✉❝❤ s♠❛❧❧❡r ♦r ♠✉❝❤ ❧❛r❣❡r t❤❛♥ ❚✱ ❛s✱ ❢♦r ❡①❛♠♣❧❡✱ tr❛♥s✐❡♥t ♣❤❡♥♦♠❡♥❛ ♦❢ ❞✉r❛t✐♦♥ ∆t << ❚ ♦r ❝②❝❧❡s t❤❛t ❡①✐st ✐♥ ♣❡r✐♦❞s ❧❛r❣❡r t❤❛♥ ❚✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚

❆ ✇❛✈❡❧❡t ψ✵(t) ✭s♦♠❡t✐♠❡s ❛❧s♦ ❝❛❧❧❡❞ ♠♦t❤❡r ✇❛✈❡❧❡t✮✱ t ∈ R✱ ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t s❛t✐s✜❡s t❤r❡❡ ❝♦♥❞✐t✐♦♥s✳

✶ ■ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ Ψ(ν)✱ −∞ < ν < ∞✱ ✐s s✉❝❤ t❤❛t ❡①✐sts ❛ ✜♥✐t❡

❝♦♥st❛♥t ❈ψ t❤❛t ♦❜❡②s t❤❡ ❛❞♠✐ss✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✵ < ❈ψ = ∞

✵

|Ψ(ν)|✷ ν ❞ν < ∞ . ✭✶✸✮

✷ ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ ψ✵(t) ✐s ♥✉❧❧✿

∞

−∞

ψ✵(t) ❞t = ✵ . ✭✶✹✮

✸ ■ts ❡♥❡r❣② ✐s ✉♥✐t❛r②✿

∞

−∞

|ψ✵(t)|✷ ❞t = ✶ . ✭✶✺✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✼ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❊①❛♠♣❧❡s ♦❢ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s

0.2 0.4 0.6 0.8 1 −0.2 −0.1 0.1 0.2 Haar Wavelet 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0.2 0.4 D4 Wavelet 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0.1 0.2 C3 Coiflet 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0.1 0.2 S8 Symmlet

❋✐❣✉r❡✿ ❋♦✉r ❡①❛♠♣❧❡s ♦❢ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✽ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❊①❛♠♣❧❡s ♦❢ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s ✲ ❝♦♥t✳

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.2 −0.1 0.1 0.2

❋✐❣✉r❡✿ ▼❡②❡r✬s ✇❛✈❡❧❡t✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✶✾ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❊①❛♠♣❧❡s ♦❢ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s ✲ ❝♦♥t✳

−5 5 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

wavelet Gaussiana de ordem 1

t

❋✐❣✉r❡✿ ●❛✉ss✐❛♥ ✇❛✈❡❧❡t ✭r❡❧❛t❡❞ t♦ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ●❛✉ss✐❛♥ P❉❋✮✳

−5 5 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2

wavelet chapéu mexicano

t

❋✐❣✉r❡✿ ✏▼❡①✐❝❛♥ ❤❛t✑ ✇❛✈❡❧❡t ✭r❡❧❛t❡❞ t♦ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ●❛✉ss✐❛♥ P❉❋✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✵ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚

❚❤❡ ✇❛✈❡❧❡t tr❛♥s❢♦r♠ ❤❛s ❜❡❡♥ ♦r✐❣✐♥❛❧❧② ❞❡✈❡❧♦♣❡❞ ❛s ❛♥ ❛♥❛❧②s✐s ❛♥❞ s②♥t❤❡s✐s t♦♦❧ ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❡♥❡r❣② s✐❣♥❛❧s✳ ❆♥ ❡♥❡r❣② s✐❣♥❛❧ ①(t)✱ t ∈ R ✭t ❞❡♥♦t❡s t✐♠❡✮✱ ♦❜❡②s t❤❡ ❝♦♥str❛✐♥t ①✷ = ①, ① ≡ ∞

−∞

|①(t)|✷ ❞t < ∞, ✭✶✻✮ ✐✳ ❡✳✱ ①(t) t❤❛t ♦❜❡②s t❤❡ ❝♦♥str❛✐♥t ✭✶✻✮ ❜❡❧♦♥❣s t♦ t❤❡ sq✉❛r❡❞ s✉♠♠❛❜❧❡ ❢✉♥❝t✐♦♥s s♣❛❝❡ ▲✷(R)✳ Pr❡s❡♥t❧②✱ t❤❡ ✇❛✈❡❧❡t tr❛♥s❢♦r♠ ❤❛s ❛❧s♦ ❜❡❡♥ ✉s❡❞ ❛s ❛♥ ❛♥❛❧②s✐s t♦♦❧ ♦❢ ❞✐s❝r❡t❡ t✐♠❡ s✐❣♥❛❧s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✶ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚ ✲ ❝♦♥t✳

❚❤❡r❡ ❛r❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ✭❈❲❚✮ ❛♥❞ ❞✐s❝r❡t❡ t✐♠❡ ✇❛✈❡❧❡t ❞❡❝♦♠♣♦s✐t✐♦♥s✳ ❚❤❡ ❈❲❚ ♦❢ ❛ s✐❣♥❛❧ ①(t) ❝♦♥s✐sts ♦❢ ❛ s❡t ❈ = {❲ψ(s, τ), s ∈ R+, τ ∈ R}✱ ✐♥ ✇❤✐❝❤

τ ✐s t❤❡ t✐♠❡ ❧♦❝❛❧✐③❛t✐♦♥ ♣❛r❛♠❡t❡r✱ s r❡♣r❡s❡♥ts s❝❛❧❡ ❛♥❞ ψ ❞❡♥♦t❡s ❛ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥

♦❢ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♦♥ t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡✲s❝❛❧❡ ♣❧❛♥❡ ✭❛❧s♦ ❦♥♦✇♥ ❛s t✐♠❡✲❢r❡q✉❡♥❝② ♣❧❛♥❡✮❣✐✈❡♥ ❜② ❲ψ(s, τ) =

• ψ✵(s,τ), ①
• =

∞

−∞

✶ √s ψ∗

✵

λ − τ s

• ①(λ)❞λ .

✭✶✼✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚ ✲ ❝♦♥t✳

ψ✵(s,τ)(t) = s−✶/✷ψ✵ t−τ

s

• ❞❡♥♦t❡s ❛ ❞✐❧❛t❡❞ ❛♥❞ s❤✐❢t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡

✏♠♦t❤❡r✑ ✇❛✈❡❧❡t ψ✵(t)✳ ❚❤❡ ❢❛❝t♦r ✶/√s ✐♥ ✭✶✼✮ ♣r♦✈✐❞❡s ❛❧❧ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❝❧❛ss W = ✶ √s ψ✵ t − τ s

• ∈ R
• ✭✶✽✮

❤❛✈❡ t❤❡ s❛♠❡ ❡♥❡r❣② ✭♥♦r♠✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲▼❚ ✲ ❝♦♥t✳

❚❤❡ ❜❛s✐❝ ✐❞❡❛ ♦❢ t❤❡ ❈❲❚ ❞❡✜♥❡❞ ❜② ✭✶✼✮ ✐s t♦ ❝♦rr❡❧❛t❡ ✶ ❛ s✐❣♥❛❧ ①(t) ✇✐t❤ s❤✐❢t❡❞ ✭❜② τ✮ ❛♥❞ ❞✐❧❛t❡❞ ✭❜② s✮ ✈❡rs✐♦♥s ♦❢ ❛ ♠♦t❤❡r ✇❛✈❡❧❡t ✭t❤❛t ❤❛s ❛ ♣❛ss✲❜❛♥❞ s♣❡❝tr✉♠✮✳ ❚❤❡ ❈❲❚ ✐s ❛ t✇♦ ♣❛r❛♠❡t❡rs ❢✉♥❝t✐♦♥✳ ❙♦✱ ✐t ✐s ❛ r❡❞✉♥❞❛♥t tr❛♥s❢♦r♠✱ ❜❡❝❛✉s❡ ✐t ❝♦♥s✐sts ♦♥ ♠❛♣♣✐♥❣ ❛♥ ♦♥❡✲❞✐♠❡♥s✐♦♥ s✐❣♥❛❧ ♦♥ t❤❡ t✐♠❡✲s❝❛❧❡ ♣❧❛♥❡✳

✶▼❡❛s✉r❡ t❤❡ ❧✐❦❡❧✐❤♦♦❞✳ ❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

■❈❲❚

❉✐✛❡r❡♥t❧② ❢r♦♠ t❤❡ ❲❋❚✱ ✇❤❡r❡ t❤❡ r❡❝♦♥str✉❝t✐♦♥ ✐s ♠❛❞❡ ❢r♦♠ t❤❡ s❛♠❡ ❢❛♠✐❧② ♦❢ ❢✉♥❝t✐♦♥s ❛s t❤❛t ✉s❡❞ ✐♥ t❤❡ ❛♥❛❧②s✐s✱ ✐♥ t❤❡ ❈❲❚ t❤❡ s②♥t❤❡s✐s ✐s ♠❛❞❡ ✇✐t❤ ❢✉♥❝t✐♦♥s ˜ ψs,τ t❤❛t ❤❛✈❡ t♦ s❛t✐s❢② ˜ ψs,τ(t) = ✶ ❈ψ ✶ s✷ ψs,τ(t) . ✭✶✾✮ ❙♦✱ ①(t) ✐s ❝♦♠♣❧❡t❡❧② r❡❝♦✈❡r❡❞ ❜② t❤❡ ✐♥✈❡rs❡ ❝♦♥t✐♥✉♦✉s ✇❛✈❡❧❡t tr❛♥s❢♦r♠ ✭■❈❲❚✮✿ ①(t) = ✶ ❈ψ ∞

✵

∞

−∞

❲ψ(s, τ) ✶ √s ψ t − τ s

• ❞τ

❞s s✷ . ✭✷✵✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚ ✈s✳ ❲❋❚

❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❈❲❚ ❛♥❞ t❤❡ ❲❋❚ ❝♦♥s✐sts ♦❢ t❤❡ ❢❛❝t t❤❛t t❤❡ ❢✉♥❝t✐♦♥s ψs,τ ✉♥❞❡r❣♦ ❞✐❧❛t✐♦♥s ❛♥❞ ❝♦♠♣r❡ss✐♦♥s✳ ❚❤❡ ❛♥❛❧②s✐s ♦♥ r❡✜♥❡❞ s❝❛❧❡s ♦❢ t✐♠❡ ✭s♠❛❧❧ ✈❛❧✉❡s ♦❢ s✮ r❡q✉✐r❡s ✏❢❛st✑ ψs,τ ❢✉♥❝t✐♦♥s✱ ✐✳ ❡✳✱ ♦❢ ❛ s♠❛❧❧ s✉♣♣♦rt✱ ✇❤✐❧❡ t❤❡ ❛♥❛❧②s✐s ♦♥ ❛❣❣r❡❣❛t❡ s❝❛❧❡s ♦❢ t✐♠❡ ✭❧❛r❣❡ ✈❛❧✉❡s ♦❢ s✮ r❡q✉✐r❡s ✏s❧♦✇❡r✑ ψs,τ ❢✉♥❝t✐♦♥s✱ ✐✳ ❡✳✱ ♦❢ ❛ ✇✐❞❡r s✉♣♣♦rt✳ ❆s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞✱ t❤❡ ✐♥t❡r♥❛❧ ♣r♦❞✉❝t ❞❡✜♥❡❞ ❜② ✭✶✼✮ ✐s ❛ ❧✐❦❡❧✐❤♦♦❞ ♠❡❛s✉r❡ ❜❡t✇❡❡♥ t❤❡ ✇❛✈❡❧❡t ψ t−τ

s

• ❛♥❞ t❤❡ s✐❣♥❛❧ ①(t)

♦♥ ❛ ❝❡rt❛✐♥ ✐♥st❛♥t ♦❢ t✐♠❡ τ ❛♥❞ ♦♥ ❛ ❞❡t❡r♠✐♥❡❞ s❝❛❧❡ s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚ ✈s ❲❋❚ ✲ ❝♦♥t✳

❋♦r ❛ ✜①❡❞ τ✱ ❧❛r❣❡ ✈❛❧✉❡s ♦❢ s ❝♦rr❡s♣♦♥❞ t♦ ❛ ❧♦✇✲❢r❡q✉❡♥❝② ❛♥❛❧②s✐s✱ ✇❤✐❧❡ s♠❛❧❧ ✈❛❧✉❡s ♦❢ s ❛r❡ ❛ss♦❝✐❛t❡❞ t♦ ❛ ❤✐❣❤✲❢r❡q✉❡♥❝② ❛♥❛❧②s✐s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✇❛✈❡❧❡t tr❛♥s❢♦r♠ ❤❛s ❛ ✈❛r✐❛❜❧❡ t✐♠❡ r❡s♦❧✉t✐♦♥ ✭✐✳ ❡✳✱ t❤❡ ❝❛♣❛❝✐t② ♦❢ ❛♥❛❧②③✐♥❣ ❛ s✐❣♥❛❧ ❢r♦♠ ❝❧♦s❡ ✲ ✏③♦♦♠ ✐♥✑ ✲ ♦r ❢r♦♠ ❢❛r ✲ ✏③♦♦♠ ♦✉t✑✮✱ ❜❡✐♥❣ ❛❞❡q✉❛t❡ t♦ ❛♥❛❧②③❡ ♣❤❡♥♦♠❡♥❛ t❤❛t ♦❝❝✉r ✐♥ ❞✐✛❡r❡♥t t✐♠❡ s❝❛❧❡s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✼ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❊①❛♠♣❧❡ ♦❢ ❈❲❚

50 100 150 200 250 −1 1 2 3

sinal x(t) tempo log2(s) CWT

50 100 150 200 250 5 10 15 20 25 30

❋✐❣✉r❡✿ ❚❤❡ ✐♠❛❣❡ ♦♥ t❤❡ ❜♦tt♦♠ ♣❛rt ♦❢ t❤❡ ✜❣✉r❡ ✐s t❤❡ ❈❲❚ ❲ψ(s, τ) ♦❢ t❤❡ s✐❣♥❛❧ ♦♥ t❤❡ t♦♣ ♣❛rt✱ ❡✈❛❧✉❛t❡❞ ✇✐t❤ ❛ ✇❛✈❡❧❡t t❤❛t ✐s t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡

• ❛✉ss✐❛♥ P❉❋✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✽ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ❛♥❞ t❤❡ ❞✐s❝r❡t❡ ✇❛✈❡❧❡t tr❛♥s❢♦r♠

❚❤❡r❡ ❛r❡ t✇♦ ❦✐♥❞s ♦❢ ❉❲❚✿

t❤❡ ❉❲❚ ❢♦r ❞✐s❝r❡t❡ t✐♠❡ s✐❣♥❛❧s ❛♥❞ t❤❡ ❉❲❚ ❢♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ s✐❣♥❛❧s✳

❚❤❡ ❉❲❚ ♠❛② ❜❡ ❢♦r♠✉❧❛t❡❞ ❢♦r ❞✐s❝r❡t❡ t✐♠❡ s✐❣♥❛❧s ✭❛s ✐t ✐s ❞♦♥❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❜② P❡r❝✐✈❛❧ ❛♥❞ ❲❛❧❞❡♥✮ ✇✐t❤♦✉t ❡st❛❜❧✐s❤✐♥❣ ❛♥② ❡①♣❧✐❝✐t ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ❈❲❚✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ s❤♦✉❧❞ ♥♦t ✉♥❞❡rst❛♥❞ t❤❡ t❡r♠ ✏❞✐s❝r❡t❡✑ ♦❢ t❤❡ ❉❲❚ ❢♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ s✐❣♥❛❧s ❛s ♠❡❛♥✐♥❣ t❤❛t t❤✐s tr❛♥s❢♦r♠ ✐s ❞❡✜♥❡❞ ♦✈❡r ❛ ❞✐s❝r❡t❡ t✐♠❡ s✐❣♥❛❧✳ ❇✉t ♦♥❧② t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ♣r♦❞✉❝❡❞ ❜② t❤✐s tr❛♥s❢♦r♠ ❜❡❧♦♥❣ t♦ ❛ s✉❜s❡t ❉ = {✇❥,❦ = ❲ψ(✷❥, ✷❥❦), ❥ ∈ Z, ❦ ∈ Z} ♦❢ t❤❡ s❡t ❈✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✷✾ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ❛♥❞ t❤❡ ❞✐s❝r❡t❡ ✇❛✈❡❧❡t tr❛♥s❢♦r♠ ✲ ❝♦♥t✳

■♥ ❢❛❝t✱ t❤❡ ❉❲❚ ❝♦❡✣❝✐❡♥ts ❢♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ s✐❣♥❛❧s ❝❛♥ ❛❧s♦ ❜❡ ❞✐r❡❝t❧② ♦❜t❛✐♥❡❞ ❜② ♠❡❛♥s ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✇❥,❦ =

• ψ✵(✷❥ ,✷❥ ❦), ①
• =

∞

−∞

✷−❥/✷ψ∗

✵(✷−❥λ − ❦)①(λ) ❞λ ,

✭✷✶✮ ✐♥ ✇❤✐❝❤ t❤❡ ✐♥❞✐❝❡s ❥ ❛♥❞ ❦ ❛r❡ ❝❛❧❧❡❞ s❝❛❧❡ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥✱ r❡s♣❡❝t✐✈❡❧②✱ ❞♦❡s ♥♦t ✐♥✈♦❧✈❡ ❛♥② ❞✐s❝r❡t❡ t✐♠❡ s✐❣♥❛❧✱ ❜✉t t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ s✐❣♥❛❧ ①(t)✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✵ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ❛♥❞ t❤❡ ❞✐s❝r❡t❡ ✇❛✈❡❧❡t tr❛♥s❢♦r♠ ✲ ❝♦♥t✳

❊q✳ ✭✷✶✮ s❤♦✇s t❤❛t t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❉❲❚ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝r✐t✐❝❛❧❧② s❛♠♣❧❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❈❲❚ ❞❡✜♥❡❞ ❜② ✭✶✼✮ ✐♥ t❤❡ ❞②❛❞✐❝ s❝❛❧❡s s = ✷❥✱ ❥ = . . . , −✶, ✵, ✶, ✷, . . .✱ ✐♥ ✇❤✐❝❤ t❤❡ ✐♥st❛♥ts ♦❢ t✐♠❡ ✐♥ t❤❡ ❞②❛❞✐❝ s❝❛❧❡ s = ✷❥ ❛r❡ s❡♣❛r❛t❡❞ ❜② ♠✉❧t✐♣❧❡s ♦❢ ✷❥✳ ❚❤❡ ❢✉♥❝t✐♦♥ ψ✵ ♦❢ ✭✷✶✮ ♠✉st ❜❡ ❞❡✜♥❡❞ ❢r♦♠ ❛ ♠✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ✭▼❘❆✮ ♦❢ t❤❡ s✐❣♥❛❧ ①(t)✳ ❖❜s❡r✈❡ t❤❛t t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ▼❘❆ t❤❡♦r② ✐s s✐♠✐❧❛r t♦ t❤❛t ♦❢ ❞✐s❝r❡t❡ t✐♠❡✳ ❆❧t❤♦✉❣❤ t❤❡ t❡❧❡tr❛✣❝ s✐❣♥❛❧s ❛r❡ ❞✐s❝r❡t❡ t✐♠❡✱ ✇❡ ❞❡❝✐❞❡❞ t♦ ♣r❡s❡♥t t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ▼❘❆ ✈❡rs✐♦♥ ❜❡❝❛✉s❡ t❤❡ ❍✉rst ♣❛r❛♠❡t❡r ❡st✐♠❛t♦r ❜❛s❡❞ ♦♥ ✇❛✈❡❧❡ts ♣r♦♣♦s❡❞ ❜② ❆❜r② ❛♥❞ ❱❡✐t❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ s♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ ❛ ✏✜❝t✐t✐♦✉s✑ ♣r♦❝❡ss {˜ ①t, t ∈ R} t❤❛t ✐s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❞✐s❝r❡t❡ t✐♠❡ ♣r♦❝❡ss {①♥, ♥ ∈ Z}✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✶ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❈❲❚ s❛♠♣❧✐♥❣

s

j = 0 j = 1 j = 2 j = 3 j = 4

❋✐❣✉r❡✿ ❈r✐t✐❝❛❧ s❛♠♣❧✐♥❣ ♦❢ t❤❡ t✐♠❡✲s❝❛❧❡ ♣❧❛♥❡ ❜② ♠❡❛♥s ♦❢ t❤❡ ❈❲❚ ♣❛r❛♠❡t❡rs ✭s = ✷❥ ❡ τ = ✷❥❦✮ ❞✐s❝r❡t✐③❛t✐♦♥✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆

❆ ▼❘❆ ✐s✱ ❜② ❞❡✜♥✐t✐♦♥✱ ❛ s❡q✉❡♥❝❡ ♦❢ ❝❧♦s❡❞ s✉❜s♣❛❝❡s {❱❥}❥∈Z ❞❡ ▲✷(R) s✉❝❤ t❤❛t✿

✶

. . . ❱✷ ⊂ ❱✶ ⊂ ❱✵ ⊂ ❱−✶ ⊂ ❱−✷ ⊂ . . .❀

✷

• ❥∈❩ ❱❥ = {}❀

✸

• ❥∈❩ ❱❥ = ▲✷(R)❀

✹

①(t) ∈ ❱❥ ⇔ ①(✷❥t) ∈ ❱✵, ❥ > ✵ ✭✐♥ ✇❤✐❝❤ t ❞❡♥♦t❡s t✐♠❡ ❛♥❞ ①(t) ✐s ❛♥ ❡♥❡r❣② s✐❣♥❛❧✮❀

✺

❚❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥ φ❥(t) = ✷−❥/✷φ✵(✷−❥t) ✐♥ ❱❥✱ ❝❛❧❧❡❞ s❝❛❧❡ ❢✉♥❝t✐♦♥✱ s✉❝❤ t❤❛t t❤❡ s❡t {φ❥,❦, ❦ ∈ Z} ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❱❥✱ ✇✐t❤ φ❥,❦(t) = ✷−❥/✷φ✵(✷−❥t − ❦) ∀❥, ❦ ∈ Z✳

❚❤❡ s✉❜s♣❛❝❡ ❱❥✐s ❦♥♦✇♥ ❛s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ t✐♠❡ s❝❛❧❡ s❥ = ✷❥ ✭❛ss✉♠✐♥❣ t❤❛t ❱✵ ✐s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡ ✇✐t❤ ✉♥✐t s❝❛❧❡✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

■❢ t❤❡ ①(t) ♣r♦❥❡❝t✐♦♥ ♦♥ ❱❥ ❞❡ ①(t) ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ s❝❛❧❡ ❝♦❡✣❝✐❡♥ts ✉❥,❦ =

• φ❥,❦, ①
• =

∞

−∞

✷−❥/✷φ∗

✵(✷−❥t − ❦)①(t) ❞t,

✭✷✷✮ t❤❡♥ t❤❡ ♣r♦♣❡rt✐❡s ✶ ❛♥❞ ✸ ❛ss✉r❡ t❤❛t ❧✐♠

❥→−∞

• ❦ φ❥,❦(t)✉❥,❦ = ①(t)✱

∀ ① ∈ ▲✷(R)✳ Pr♦♣❡rt② ✹ ✐♠♣❧✐❡s t❤❛t t❤❡ s✉❜s♣❛❝❡ ❱❥ ✐s ❛ s❝❛❧❡❞ ✈❡rs✐♦♥ ♦❢ s✉❜s♣❛❝❡ ❱✵ ✭♠✉❧t✐r❡s♦❧✉t✐♦♥✮✳ ❚❤❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♠❡♥t✐♦♥❡❞ ✐♥ ♣r♦♣❡rt② ✺ ✐s ♦❜t❛✐♥❡❞ ❜② t✐♠❡ s❤✐❢ts ♦❢ t❤❡ ❧♦✇✲♣❛ss ❢✉♥❝t✐♦♥ φ❥✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❈♦♥s✐❞❡r t❤❡ s✉❝❝❡ss✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥s s❡q✉❡♥❝❡ ✭❛❧s♦ ❦♥♦✇♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛s ✇❛✈❡❧❡t s♠♦♦t❤s✮ ♦❢ ①(t) S❥(t) =

• ❦

φ❥,❦(t)✉❥,❦ ❥ = . . . , −✶, ✵, ✶, . . . . ✭✷✸✮ ❆s ❱❥+✶ ⊂ ❱❥✱ ✇❡ ❤❛✈❡ S❥+✶(t) ✐s ❛ ❝♦❛rs❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ①(t) t❤❛♥ S❥(t)✳ ❚❤✐s ❢❛❝t ✐❧❧✉str❛t❡s t❤❡ ▼❘❆✬s ❢✉♥❞❛♠❡♥t❛❧ ✐❞❡❛✱ t❤❛t ❝♦♥s✐sts ✐♥ ❡①❛♠✐♥✐♥❣ t❤❡ ❧♦ss ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✇❤❡♥ ♦♥❡ ❣♦❡s ❢r♦♠ S❥(t) t♦ S❥+✶(t)✿ S❥(t) = S❥+✶(t) + ∆①❥+✶(t). ✭✷✹✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

∆①❥+✶(t) ✭❝❛❧❧❡❞ ❞❡t❛✐❧ ♦❢ ①❥(t)✮ ❜❡❧♦♥❣s t♦ t❤❡ s✉❜s♣❛❝❡ ❲❥+✶✱ ♥❛♠❡❞ ❞❡t❛✐❧ s♣❛❝❡ t❤❛t ✐s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✢✉❝t✉❛t✐♦♥s ✭♦r ✈❛r✐❛t✐♦♥s✮ ♦❢ t❤❡ s✐❣♥❛❧ ✐♥ t❤❡ ♠♦r❡ r❡✜♥❡❞ t✐♠❡ s❝❛❧❡ s❥ = ✷❥ ❛♥❞ t❤❛t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♦rt❤♦❣♦♥❛❧ ❝♦♠♣❧❡♠❡♥t ♦❢ ❱❥+✶ ✐♥ ❱❥ ✷✳ ❚❤❡ ▼❘❆ s❤♦✇s t❤❛t t❤❡ ❞❡t❛✐❧ s✐❣♥❛❧s ∆①❥+✶(t) = D❥+✶(t) ♠❛② ❜❡ ❞✐r❡❝t❧② ♦❜t❛✐♥❡❞ ❜② s✉❝❝❡ss✐✈❡ ♣r♦❥❡❝t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s✐❣♥❛❧ ①(t) ♦♥ ✇❛✈❡❧❡t s✉❜s♣❛❝❡s ❲❥✳ ❇❡s✐❞❡s✱ t❤❡ ▼❘❆ t❤❡♦r② s❤♦✇s t❤❛t ❡①✐sts ❛ ❢✉♥❝t✐♦♥ ψ✵(t)✱ ❝❛❧❧❡❞ ✏♠♦t❤❡r ✇❛✈❡❧❡t✑ ✱ t❤❛t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ φ✵(t)✱ ✐♥ ✇❤✐❝❤ ψ❥,❦(t) = ✷−❥/✷φ✵(✷−❥t − ❦) ❦ ∈ Z ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❲❥✳

✷❇❡s✐❞❡s✱ ❲❥+✶ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ s✉❜s♣❛❝❡ ❱❥✳ ❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❚❤❡ ❞❡t❛✐❧ D❥+✶(t) ✐s ♦❜t❛✐♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥ D❥+✶(t) =

• ❦

ψ❥+✶,❦(t)

• ψ❥+✶,❦(t), ①(t)
• .

✭✷✺✮ ❚❤❡ ✐♥t❡r♥❛❧ ♣r♦❞✉❝t

• ψ❥+✶,❦(t), ①(t)
• = ✇❥+✶,❦ ❞❡♥♦t❡s t❤❡ ✇❛✈❡❧❡t

❝♦❡✣❝✐❡♥t ❛ss♦❝✐❛t❡❞ t♦ s❝❛❧❡ ❥ + ✶ ❛♥❞ ❞✐s❝r❡t❡ t✐♠❡ ❦ ❛♥❞ {ψ❥+✶,❦(t)} ✐s ❛ ❢❛♠✐❧② ♦❢ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s t❤❛t ❣❡♥❡r❛t❡s t❤❡ s✉❜s♣❛❝❡ ❲❥+✶✱ ♦rt❤♦❣♦♥❛❧ t♦ s✉❜s♣❛❝❡ ❱❥+✶ ✭❲❥+✶⊥❱❥+✶✮✱ ✐✳ ❡✳✱ ψ❥+✶,♥, φ❥+✶,♣ = ✵ , ∀♥, ♣. ✭✷✻✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✼ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❚❤❡r❡❢♦r❡✱ t❤❡ ❞❡t❛✐❧ s✐❣♥❛❧ D❥+✶(t) ❜❡❧♦♥❣s t♦ t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② s✉❜s♣❛❝❡ ❲❥+✶ ❞❡ ❱❥✱ ❜❡❝❛✉s❡ ❱❥ = ❱❥+✶ ⊕ ❲❥+✶. ✭✷✼✮ ❚❤❛t ✐s✱ ❱❥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❞✐r❡❝t ❛❞❞✐t✐♦♥ ♦❢ ❱❥+✶ ❛♥❞ ❲❥+✶✱ ❛♥❞ t❤✐s ♠❡❛♥s t❤❛t ❛♥② ❡❧❡♠❡♥t ✐♥ ❱❥ ♠❛② ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ t✇♦ ♦rt❤♦❣♦♥❛❧ ❡❧❡♠❡♥ts ❜❡❧♦♥❣✐♥❣ t♦ ❱❥+✶ ❛♥❞ ❲❥+✶✳ ■t❡r❛t✐♥❣ ✭✷✼✮✱ ✇❡ ❤❛✈❡ ❱❥ = ❲❥+✶ ⊕ ❲❥+✷ ⊕ . . . . ✭✷✽✮ ❊q✳ ✭✷✽✮ s❛②s t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ S❥(t) ✐s ❣✐✈❡♥ ❜② S❥(t) =

∞

• ✐=❥+✶
• ❦

✇✐,❦ψ✐,❦(t) . ✭✷✾✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✽ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❚❤❡ ▼❘❆ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ s✐❣♥❛❧ ①(t) ✐s ✐♥✐t✐❛t❡❞ ❜② ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts✸ ✉✵(❦) =

• φ✵,❦(t), ①(t)
• ✱ ✐♥ ✇❤✐❝❤ ❦ = ✵, ✶, . . . , ◆ − ✶✱

t❤❛t ❛r❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ①(t) ♦♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s✉❜s♣❛❝❡ ❱✵✳

✸❚❤❡ s❡q✉❡♥❝❡ ✉✵(❦) ✐s ♦❜t❛✐♥❡❞ s❛♠♣❧✐♥❣ t❤❡ ✜❧t❡r✬s ♦✉t♣✉t ✇❤♦s❡ ✐♠♣✉❧s❡ r❡s♣♦♥s❡

✐s φ∗(−t) ✭♠❛t❝❤❡❞ ✜❧t❡r ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ φ✵(t) = φ(t)✮ ❛t ✐♥st❛♥ts ❦ = ✵, ✶, ✷, . . .✱ ✐✳ ❡✳✱ ✉✵(❦) = ①(t) ⋆ φ∗(−t) ❢♦r ❦ = ✵, ✶, ✷, . . .✱ ✐♥ ✇❤✐❝❤ ⋆ ❞❡♥♦t❡s ❝♦♥✈♦❧✉t✐♦♥✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✸✾ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❋♦❧❧♦✇✐♥❣✱ t❤❡ s❡q✉❡♥❝❡ {✉✵(❦)} ✐s ❞❡❝♦♠♣♦s❡❞ ❜② ✜❧t❡r✐♥❣ ❛♥❞ s✉❜✲s❛♠♣❧✐♥❣ ❜② ❛ ❢❛❝t♦r ♦❢ ✷ ✭❞♦✇♥s❛♠♣❧✐♥❣✮ ✐♥ t✇♦ s❡q✉❡♥❝❡s✿ {✉✶(❦)} ❛♥❞ {✇✶(❦)}✱ ❡❛❝❤ ♦♥❡ ✇✐t❤ ◆/✷ ♣♦✐♥ts✳ ❚❤✐s ✜❧t❡r✐♥❣ ❛♥❞ s✉❜✲s❛♠♣❧✐♥❣ ♣r♦❝❡ss ✐s r❡♣❡❛t❡❞ s❡✈❡r❛❧ t✐♠❡s✱ ♣r♦❞✉❝✐♥❣ t❤❡ s❡q✉❡♥❝❡s {{✉✵(❦)}◆, {✉✶(❦)} ◆

✷ , {✉✷(❦)} ◆ ✹ , . . . , {✉❥(❦)} ◆ ✷❥ , . . . , {✉❏(❦)} ◆ ✷❏ }

✭✸✵✮ ❛♥❞ {{✇✶(❦)} ◆

✷ , {✇✷(❦)} ◆ ✹ , . . . , {✇❥(❦)} ◆ ✷❥ , {✇❏(❦)} ◆ ✷❏ } .

✭✸✶✮ ❚❤❡ ❧✐t❡r❛t✉r❡ ❝❛❧❧s t❤❡ s❡t ♦❢ ❝♦❡✣❝✐❡♥ts

• {✇✶(❦)} ◆

✷ , {✇✷(❦)} ◆ ✹ , . . . , {✇❏(❦)} ◆ ✷❏ , {✉❏(❦)} ◆ ✷❏

• ✭✸✷✮

❛s t❤❡ ❉❲❚ ♦❢ t❤❡ ①(t) s✐❣♥❛❧✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✵ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❋✐❣✳ ✽ ✐❧❧✉str❛t❡s ❛ ✸✲❧❡✈❡❧s ❉❲❚ ✭❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ s❝❛❧❡s ❥ = ✶, ✷, ✸✮ ❛ss♦❝✐❛t❡❞ t♦ ✶✵✷✹ s❛♠♣❧❡s ♦❢ t❤❡ ❞✐s❝r❡t❡ t✐♠❡ ①(❦) = s✐♥ (✸❦) + s✐♥ (✵.✸❦) + s✐♥ (✵.✵✸❦)✱ t❤❛t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ s✉♣❡r♣♦s✐t✐♦♥ ♦❢ ✸ s✐♥✉s♦✐❞s ✐♥ ❢r❡q✉❡♥❝✐❡s ❢✶ ≈ ✵.✵✵✹✼✼✺✱ ❢✷ ≈ ✵.✵✹✼✼✺ ❛♥❞ ❢✸ ≈ ✵.✹✼✼✺✳ ❋✐❣✳ ✾ s❤♦✇s t❤❡ ❙❉❋ ♦❢ t❤✐s s✐❣♥❛❧✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✶ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❉❲❚ ❡①❛♠♣❧❡

100 200 300 400 500 600 700 800 900 1000 −4 −2 2 4

sinal original

100 200 300 400 500 600 700 800 900 1000 −6 −4 −2 2 4 6

DWT

❋✐❣✉r❡✿ ❆♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ✸✲❧❡✈❡❧s ❉❲❚ ♦❢ t❤❡ ❞✐s❝r❡t❡ t✐♠❡ s✐❣♥❛❧ ①(❦) = s✐♥ (✸❦) + s✐♥ (✵.✸❦) + s✐♥ (✵.✵✸❦)✳ ❚❤❡ ❣r❛♣❤ ❝♦♥❝❛t❡♥❛t❡s t❤❡ s❡q✉❡♥❝❡s ♦❢ t❤❡ s❝❛❧❡ ❝♦❡✣❝✐❡♥ts {✉✸(❦)}✶✷✽ ❛♥❞ ♦❢ t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts {✇✸(❦)}✶✷✽✱ {✇✷(❦)}✷✺✻ ❡ {✇✶(❦)}✺✶✷ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ✐✳ ❡✳✱ t❤❡ ✜rst ✶✷✽ ♣♦✐♥ts ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s❡q✉❡♥❝❡ {✉✸(❦)}✶✷✽❀ t❤❡♥ ❢♦❧❧♦✇ t❤❡ ✶✷✽ ♣♦✐♥ts ♦❢ t❤❡ s❡q✉❡♥❝❡ {✇✸(❦)}✶✷✽✱ t❤❡ ✷✺✻ ♣♦✐♥ts ♦❢ t❤❡ s❡q✉❡♥❝❡ {✇✷(❦)}✷✺✻ ❛♥❞ ✺✶✷ ♣♦✐♥ts ♦❢ t❤❡ s❡q✉❡♥❝❡ {✇✶(❦)}✺✶✷✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −50 −40 −30 −20 −10 10 20

Frequency (Hz)

Power/frequency (dB/Hz)

DEP

❋✐❣✉r❡✿ ❙❉❋ ♦❢ t❤❡ s✐❣♥❛❧ ①(❦) = s✐♥ (✸❦) + s✐♥ (✵.✸❦) + s✐♥ (✵.✵✸❦)✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

▼❘❆ ✲ ❝♦♥t✳

❚❤❡ r❡❝♦♥str✉❝t✐♦♥ ♦❢ ①(t) ✐s ✐♠♣❧❡♠❡♥t❡❞ ❜② ✜❧t❡r✐♥❣ ❛♥❞ ♦✈❡rs❛♠♣❧✐♥❣ ❜② ❛ ❢❛❝t♦r ♦❢ ✷ ✭✉♣s❛♠♣❧✐♥❣✮ ♦❢ t❤❡ s❡q✉❡♥❝❡s ✭✸✵✮ ❛♥❞ ✭✸✶✮✱ ♦❜t❛✐♥✐♥❣ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ①(t) ✐♥ t❤❡ s✉❜s♣❛❝❡ ❱✵ S✵(t) = S❏(t) + D✶(t) + D✷(t) + · · · + D❏(t) ✭✸✸✮ ♦r ①(t) ≈

• ❦

✉(❏, ❦)φ❏,❦(t) +

❏

• ❥=✶
• ❦

✇❥,❦ψ❥,❦(t) . ✭✸✹✮ ❊q✳ ✭✸✹✮ ❞❡✜♥❡s t❤❡ ■♥✈❡rs❡ ❉✐s❝r❡t❡ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ✲ ■❉❲❚✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❋✐❣✉r❡✿ ❙②♥t❤❡s✐s ♦❢ t❤❡ s✐❣♥❛❧ ①(❦) = s✐♥ (✸❦) + s✐♥ (✵.✸❦) + s✐♥ (✵.✵✸❦) ✐♥ t❡r♠s ♦❢ t❤❡ s✉♠ S✸(t) + D✶(t) + D✷(t) + D✸(t)✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❲❡ s❛② t❤❛t t❤❡ ❢✉♥❝t✐♦♥ φ✵(t) = φ(t) ❞❡t❡r♠✐♥❡s ❛ ▼❘❆ ♦❢ ①(t) ❛❝❝♦r❞✐♥❣ t♦ ✭✸✸✮✱ ✐❢ ✐t ♦❜❡②s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿

✶ ✐♥tr❛✲s❝❛❧❡ ♦rt❤♦♥♦r♠❛❧✐t② ✭♣r♦♣❡rt② ✺✮

φ(t − ♠), φ(t − ♥) = δ♠,♥ , ✭✸✺✮ ✐♥ ✇❤✐❝❤ δ♠,♥ ✐s t❤❡ ❑r♦♥❡❝❦❡r✬s ❞❡❧t❛ ✭δ♠,♥ = ✶ ✐❢ ♠ = ♥✱ δ♠,♥ = ✵ ❢♦r ♠ = ♥✮✳ ❊q✳ ✭✸✺✮ ✐♠♣♦s❡s ❛♥ ♦rt❤♦♥♦r♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ❛t s❝❛❧❡ ❥ = ✵✳

✷ ✉♥✐t ♠❡❛♥

∞

−∞

φ(t) ❞t = ✶ . ✭✸✻✮

✸

✶ √ ✷ φ(t ✷) =

• ♥

❣♥φ(t − ♥) , ✭✸✼✮ ❛s s❡✈❡r❛❧ φ(t − ❦) ✜t ✐♥ φ( t

✷) ✭✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ♣r♦♣❡rt② ✭✶✮ ♦❢ t❤❡

▼❘❆✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❊q✳ ✸✼ ♠❛② ❜❡ r❡✇r✐tt❡♥ ❛s φ(t) =

• ♥

√ ✷❣♥φ(✷t − ♥) , ✭✸✽✮ ❦♥♦✇♥ ❛s ❉✐❧❛t✐♦♥ ❊q✉❛t✐♦♥✳ ❊qs✳ ✸✼ ❛♥❞ ✸✽ ♠❛② ❜❡ ✇r✐tt❡♥✱ r❡s♣❡❝t✐✈❡❧②✱ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ❛s √ ✷Φ(✷ν) = ●(ν)Φ(ν) , ✭✸✾✮ ❛♥❞ Φ(ν) = ✶ √ ✷

• (ν)Φ(ν

✷) , ✭✹✵✮ ✐♥ ✇❤✐❝❤ Φ(ν) ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ φ(t) ❛♥❞

• (ν) =

♥ ❣♥❡−❥✷πν♥✱ ❦♥♦✇♥ ❛s s❝❛❧❡ ✜❧t❡r ✭❧♦✇✲♣❛ss✮✱ r❡♣r❡s❡♥ts ❛

♣❡r✐♦❞✐❝ ✜❧t❡r ✐♥ ν✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✼ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❆s t❤❡ s✉❜s♣❛❝❡ ❲❥+✶ ✐s ♦rt❤♦❣♦♥❛❧ t♦ ❱❥+✶ ❛♥❞ ✐s ✐♥ ❱❥✱ ✇❡ ❤❛✈❡ ✶ √ ✷ ψ(t ✷) =

• ♥

❤♥φ(t − ♥) , ✭✹✶✮ ♦r ψ(t) =

• ♥

√ ✷❤♥φ(✷t − ♥) , ✭✹✷✮ t❤❛t ✐s t❤❡ ❲❛✈❡❧❡t ❊q✉❛t✐♦♥✳ ❆♣♣❧②✐♥❣ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ t♦ ✭✹✶✮ ❛♥❞ ✭✹✷✮ ✇❡ ❣❡t✱ r❡s♣❡❝t✐✈❡❧②✱

• (✷)Ψ(✷ν) = ❍(ν)Φ(ν) ,

✭✹✸✮ ❛♥❞ Ψ(ν) = ✶ √ ✷ ❍(ν)Φ(ν ✷) . ✭✹✹✮ ✐♥ ✇❤✐❝❤ ❍(ν) ✐s t❤❡ ✇❛✈❡❧❡t ✜❧t❡r ✭❤✐❣❤✲♣❛ss✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✽ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❘❡✇r✐t✐♥❣ ✭✷✻✮ ✐♥ t❡r♠s ♦❢ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ❛♥❞ ✉s✐♥❣ ✭✸✾✮ ❛♥❞ ✭✹✸✮ r❡s✉❧ts t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥ ∞

−∞

• (ν)❍∗(ν)|Φ(ν)|✷ ❞ν = ✵ ,

✭✹✺✮ t❤❛t t❤❡ ✜❧t❡r ❍ ❤❛s t♦ ♦❜❡② s♦ t❤❡ ❢❛♠✐❧② {ψ✶,❦(t)} ✐s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❢❛♠✐❧② {φ✶,❦(t)}✳ ❲❡ ♠❛② s❤♦✇ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ❤♥ = (−✶)♥❣▲−✶−♥ , ↔ ❍(③) = −③−▲+✶●(−③−✶) , ✭✹✻✮ ✐♥ ✇❤✐❝❤ ▲ ❞❡♥♦t❡s t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❋■❘ ✜❧t❡r ❣♥✱ ✐s s✉✣❝✐❡♥t t♦ ✭✹✺✮ t♦ ❤♦❧❞✳ ❲❡ s❛② t❤❛t ❣♥ ❡ ❤♥ ❛r❡ q✉❛❞r❛t✉r❡ ♠✐rr♦r❡❞ ✜❧t❡rs ✭♦r ◗▼❋✮ ✇❤❡♥ t❤❡② ❛r❡ r❡❧❛t❡❞ ❜② ✭✹✻✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✹✾ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

f |H(f)| |G(f)| QMF f |H(f)| |G(f)| Filtros brickwall

❋✐❣✉r❡✿ ◗▼❋ ✜❧t❡rs ❢r❡q✉❡♥❝② r❡s♣♦♥s❡ ✭❣r❛♣❤s ♦♥ t❤❡ ✉♣♣❡r ♣❛rt✮ ✈s ❜r✐❝❦✇❛❧❧✲t②♣❡ ✜❧t❡rs ❢r❡q✉❡♥❝② r❡s♣♦♥s❡✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✵ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❆❝❝♦r❞✐♥❣ t♦ ✭✸✽✮✱ t❤❡ ▼❘❆ ❞❡♣❛rts ❢r♦♠ ❛ ❞❡✜♥✐t✐♦♥ ✭❢r♦♠ s❡✈❡r❛❧ ♣♦ss✐❜❧❡✮ ♦❢ t❤❡ s❝❛❧❡ ❢✉♥❝t✐♦♥ φ(t)✱ t❤❛t ✐s r❡❧❛t❡❞ t♦ t❤❡ s❝❛❧❡ ✜❧t❡r ❣♥ ❜② ✭✸✼✮✳ ❊q✳ ✭✹✻✮ s❛②s t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ❋■❘✲t②♣❡ ✜❧t❡r {❣♥} ✐♠♣❧✐❡s ❛ {❤♥} t❤❛t ✐s ❛❧s♦ ❋■❘✳ ❆t ❧❛st✱ t❤❡ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭✹✶✮✳ ❚❤❡ s❝❛❧❡ φ(t) ❛♥❞ ✇❛✈❡❧❡t ψ(t) ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❋■❘ ✜❧t❡rs {❣♥} ❡ {❤♥} ❤❛✈❡ ❝♦♠♣❛❝t s✉♣♣♦rt✱ t❤✉s ♦✛❡r✐♥❣ t❤❡ t✐♠❡ r❡s♦❧✉t✐♦♥ ❢✉♥❝t✐♦♥❛❧✐t②✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✶ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❚❤❡ s✐♠♣❧❡st s❝❛❧❡ ❢✉♥❝t✐♦♥ t❤❛t s❛t✐s✜❡s ✭✸✺✮ ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ■ = [✵, ✶)✱ t❤❛t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❍❛❛r✬s s❝❛❧❡ ❢✉♥❝t✐♦♥✿ φ(❍)(t) = χ[✵,✶)(t) =

• ✶ s❡ ✵ ≤ t < ✶

✵ ♦t❤❡r✇✐s❡. ✭✹✼✮ ■♥ t❤✐s ❝❛s❡ ✭❍❛❛r ▼❘❆✮✱ t❤❡ ❛ss♦❝✐❛t❡❞ ❍❛❛r s❝❛❧❡ ✜❧t❡r ✐s ❣✐✈❡♥ ❜② ❣♥ = {. . . , ✵, ❣✵ = ✶/ √ ✷, ❣✶ = ✶/ √ ✷, ✵, . . .} , ✭✹✽✮ t❤❡ ❍❛❛r ✇❛✈❡❧❡t ✜❧t❡r ❜② ❤♥ = {. . . , ✵, ❤✵ = ❣✶ = ✶/ √ ✷, ❤✶ = −❣✵ = −✶/ √ ✷, ✵, . . .} ✭✹✾✮ ❛♥❞ t❤❡ ❍❛❛r ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥ ❜② ψ(❍)(t) = χ[✵,✶/✷)(t) − χ[✶/✷,✶)(t) . ✭✺✵✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❋✐❣✳ ✶✷ s❤♦✇s t❤❡ ❉❛✉❜❡❝❤✐❡s✬ s❝❛❧❡ ❛♥❞ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s ✇✐t❤ ◆ = ✷, ✸, ✹ ✈❛♥✐s❤✐♥❣ ♠♦♠❡♥ts ∞

−∞

t♠ψ(t) ❞t = ✵, ♠ = ✵, ✶, . . . , ◆ − ✶ . ✭✺✶✮ ■♥❣r✐❞ ❉❛✉❜❡❝❤✐❡s ✇❛s t❤❡ ✜rst ♦♥❡ t♦ ♣r♦♣♦s❡ ❛ ♠❡t❤♦❞ ❢♦r ❜✉✐❧❞✐♥❣ s❡q✉❡♥❝❡s ♦❢ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s {● (◆)(③)}◆=✶,✷,✸,... ❛♥❞ {❍(◆)(③)}◆=✶,✷,✸,...✱ ✐♥ ✇❤✐❝❤ ● (◆)(③) ✐s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❧♦✇✲♣❛ss ❋■❘ ✜❧t❡r ❣(◆)

♥

❛♥❞ ❍(◆)(③) t♦ t❤❡ ❤✐❣❤✲♣❛ss ✜❧t❡r ❤(◆)

♥

✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❝❛❧❡ ❛♥❞ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s ❤❛✈❡ s✉♣♣♦rt ✐♥ [✵, ✷◆ − ✶]✳ ❚❤❡ ✜rst ♠❡♠❜❡r ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐s t❤❡ ❍❛❛r s②st❡♠ φ(✶) = φ(❍)✱ ψ(✶) = ψ(❍)✳ ❚❤❡ ❉❛✉❜❡❝❤✐❡s✬ ✜❧t❡rs ❛r❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ❍❛❛r s②st❡♠ ❢♦r ◆ ≥ ✷✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 1 2 3 4 5 −0.5 0.5 1 1.5 1 2 3 4 5 6 7 −0.5 0.5 1 1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 2 −3 −2 −1 1 2 3 4 −1 −0.5 0.5 1 1.5

❋✐❣✉r❡✿ ❚❤❡ ❣r❛♣❤s ✐♥ t❤❡ ❧♦✇❡r ♣❛rt s❤♦✇ t❤❡ ❉❛✉❜❡❝❤✐❡s✬ ✇❛✈❡❧❡ts ✇✐t❤ ◆ = ✷, ✸, ✹ ✈❛♥✐s❤✐♥❣ ♠♦♠❡♥ts✱ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❝❛❧❡ ❢✉♥❝t✐♦♥s ❛r❡ ✐♥ t❤❡ ✉♣♣❡r ♣❛rt✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❲❡ ❝❛♥ ❞❡♠♦♥str❛t❡ t❤❛t✿ ✉❥(♥) =

• ❦

❣(❦ − ✷♥)✉❥−✶(❦) ✭✺✷✮ ❛♥❞ t❤❛t ✇❥(♥) =

• ❦

❤(❦ − ✷♥)✉❥−✶(❦) . ✭✺✸✮ ❆❝❝♦r❞✐♥❣ t♦ ✭✺✷✮ ❛♥❞ ✭✺✸✮✱ ✇❡ ❝❛♥ ♦❜t❛✐♥ t❤❡ ❝♦❡✣❝✐❡♥ts ✉❥(♥) ❛♥❞ ✇❥(♥) ❢r♦♠ t❤❡ s❝❛❧❡ ❝♦❡✣❝✐❡♥ts ✉❥−✶(♠) ❜② ♠❡❛♥s ♦❢ ❞❡❝✐♠❛t✐♦♥ ♦♣❡r❛t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡ {✉❥−✶(♠)} ❜② ❛ ❢❛❝t♦r ♦❢ ✷✳ ❚❤❡ ❞❡❝✐♠❛t✐♦♥ ❝♦♥s✐sts ✐♥ ❝❛s❝❛❞✐♥❣ ❛ ❧♦✇✲♣❛ss ✜❧t❡r ❣(−♠) ✭✇✐t❤ ❛ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ¯

• (③) = ●(✶/③) ❛♥❞ ❢r❡q✉❡♥❝② r❡s♣♦♥s❡ ● ∗(❢ )✮ ♦r ❛

❤✐❣❤✲♣❛ss ❤(−♠) ✭✇✐t❤ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ¯ ❍(③) = ❍(✶/③) ❛♥❞ ❢r❡q✉❡♥❝② r❡s♣♦♥s❡ ❍∗(❢ )✮ ✇✐t❤ ❛ ❝♦♠♣r❡ss♦r ✭♦r ❞❡❝✐♠❛t♦r✮ ❜② ❛ ❢❛❝t♦r ♦❢ ✷✳ ❉❡❝✐♠❛t❡ ❛ s✐❣♥❛❧ ❜② ❛ ❢❛❝t♦r ❉ ✐s t❤❡ s❛♠❡ ❛s t♦ r❡❞✉❝❡ ✐ts s❛♠♣❧✐♥❣ r❛t❡ ❜② ❉ t✐♠❡s✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

❚❤❡ ▼❘❆ ✐s ✐♠♣❧❡♠❡♥t❡❞ ❜② ❛ ❧♦✇✲♣❛ss ❛♥❞ ❤✐❣❤✲♣❛ss ❛♥❛❧②s✐s ✜❧t❡r ❜❛♥❦s ● ∗(❢ ) ❛♥❞ ❍∗(❢ ) ❛❞❡q✉❛t❡❧② ♣♦s✐t✐♦♥❡❞ ❢♦r s❡♣❛r❛t✐♥❣ t❤❡ s❝❛❧❡ ❛♥❞ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts s❡q✉❡♥❝❡s✳ ▲❛t❡r✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ r❡❜✉✐❧❞ t❤❡ ♦r✐❣✐♥❛❧ s✐❣♥❛❧ ✉s✐♥❣ ❞✉❛❧ ◗▼❋ r❡❝♦♥str✉❝t✐♦♥ ✜❧t❡r ❜❛♥❦s✱ ❧♦✇✲♣❛ss ●(❢ ) ❛♥❞ ❤✐❣❤✲♣❛ss ❍(❢ )✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ❡♠♣❤❛s✐③❡ t❤❛t t❤❡ ♣②r❛♠✐❞ ❛❧❣♦r✐t❤♠✬s ❝♦♠♣❧❡①✐t② ✐s ❖(◆) ✭❛ss✉♠✐♥❣ ✇❡ ✇❛♥t t♦ ❡✈❛❧✉❛t❡ t❤❡ ❉❲❚ ♦❢ ◆ s❛♠♣❧❡s✮✱ ✇❤✐❧❡ t❤❡ ❞✐r❡❝t ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❉❲❚ ✭t❤❛t ✐♥✈♦❧✈❡s ♠❛tr✐❝❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥✮ ✐s ❖(◆✷)✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

g(-m) G*(f) h(-m) H*(f)

uj-1(m)

2 2

uj(n) wj(n) QMF

❋✐❣✉r❡✿ ◗▼❋ ❛♥❛❧②s✐s ✜❧t❡r ❜❛♥❦s ● ∗(❢ ) ✭❧♦✇✲♣❛ss✮ ❛♥❞ ❍∗(❢ ) ✭❤✐❣❤✲♣❛ss✮ ✇✐t❤ ❞❡❝✐♠❛t✐♦♥ ✭❞♦✇♥s❛♠♣❧✐♥❣✮ ❜② ❛ ❢❛❝t♦r ♦❢ ✷

uj(m)

2

uj-1(n) wj(m) QMF dual

2

+

uup

j(n)

g(n) G(f) h(n) H(f) wup

j(n)

❋✐❣✉r❡✿ ◗▼❋ r❡❝♦♥str✉❝t✐♦♥ ✜❧t❡r ❜❛♥❦s ✇✐t❤ ✐♥t❡r♣♦❧❛t✐♦♥ ✭✉♣s❛♠♣❧✐♥❣✮ ❜② ❛ ❢❛❝t♦r ♦❢ ✷✳ ❖❜s❡r✈❡ t❤❛t ❛r❡ ✉s❡❞ ❞✉❛❧ ❧♦✇✲♣❛ss ❛♥❞ ❤✐❣❤✲♣❛ss ✜❧t❡rs✱ ●(❢ ) ❛♥❞ ❍(❢ )✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✼ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

t

x(t) V0

2

N amostras

w1(n) W1

N/2 amostras

G* k/N]

u0(n)

H* k/N]

G* k/(N/2)]

2 2 2

u1(n) V1 u2(n) V2 u2(n) W2

H* k/(N/2)] N/4 amostras N/2 amostras N/4 amostras

❋✐❣✉r❡✿ ❋❧♦✇ ❞✐❛❣r❛♠ t❤❛t s❤♦✇s t❤❡ ✐♥✐t✐❛❧ ♣r♦❥❡❝t✐♦♥ ♦❢ ❛ s✐❣♥❛❧ ①(t) ♦♥ ❱✵ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ❲✶✱ ❲✷ ❛♥❞ ❱✷✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✽ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

V2 u0(n) u2(n)

N/2 amostras

t

W2 w2(n) u1(n)

N/4 amostras

+

2

H k/(N/2)]

2

+

w1(n)

N amostras

x(t)

G k/(N/2)]

2

H k/N]

2

G k/N] D/A

❋✐❣✉r❡✿ ❋❧♦✇ ❞✐❛❣r❛♠ t❤❛t ✐❧❧✉str❛t❡s t❤❡ ❛♣♣r♦①✐♠❛t❡ s②♥t❤❡s✐s ♦❢ ①(t) ❢r♦♠ ❲✶✱ ❲✷ ❛♥❞ ❱✷✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✺✾ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠

f W1 W2 U2

❋✐❣✉r❡✿ ❇❧♦❝❦ ❞✐❛❣r❛♠ t❤❛t s❤♦✇s t❤❛t t❤❡ ❉❲❚ ✇♦r❦s ❛s ❛ s✉❜✲❜❛♥❞s ❝♦❞✐✜❝❛t✐♦♥ s❝❤❡♠❡✳ ❚❤❡ s♣❡❝tr✉♠ ❯✵(❢ ) ♦❢ t❤❡ s✐❣♥❛❧ ✉✵(♥) ✐s s✉❜❞✐✈✐❞❡❞ ✐♥ t❤r❡❡ ❢r❡q✉❡♥❝② ❜❛♥❞s ✭t❤❛t ❝♦✈❡r t✇♦ ♦❝t❛✈❡s✮✿ ✵ ≤ ❢ < ✶/✽✱ ✶/✽ ≤ ❢ < ✶/✹ ❛♥❞ ✶/✹ ≤ ❢ ≤ ✶✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✵ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❖✉t❧✐♥❡

✶

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ◆♦♥✲♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣ ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ▼♦❞❡❧♦ ▼❲▼

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✶ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❚❤❡ ▼❲▼ ✉s❡s t❤❡ ❍❛❛r✬s ▼❘❆ ❛♥❞ ✐s ❜❛s❡❞ ♦♥ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❜✐♥♦♠✐❛❧ ❝❛s❝❛❞❡ ✐♥ t❤❡ ✇❛✈❡❧❡t ❞♦♠❛✐♥✱ t❤❛t ❡♥s✉r❡s t❤❛t t❤❡ s✐♠✉❧❛t❡❞ s❡r✐❡s ❛r❡ ♣♦s✐t✐✈❡✳ ❚❤❡ ❜✐♥♦♠✐❛❧ ❝❛s❝❛❞❡ ✐s ❛ r❛♥❞♦♠ ❜✐♥♦♠✐❛❧ tr❡❡ ✇❤♦s❡ r♦♦t ✐s t❤❡ ❝♦❡✣❝✐❡♥t ✉❏−✶,✵ ✭t❤❡ ▼❲▼ ❝♦♥s✐❞❡rs t❤❛t

◆ ✷❏−✶ = ✶✱ ✇❤❡r❡ ◆ ✐s t❤❡

♥✉♠❜❡r ♦❢ s❛♠♣❧❡s✮ ①t = ①✵,❦ = ✉❏−✶,✵φ❏−✶,✵(t) +

❏−✶

• ❥=✶
• ❦

✇❥,❦ψ❥,❦(t), ✭✺✹✮ ✐♥ ✇❤✐❝❤ φ❏−✶,✵(t) ❞❡♥♦t❡s ❛ ❍❛❛r✬s s❝❛❧❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ s❧♦✇❡st s❝❛❧❡ ✭♦r❞❡♠ ❏ − ✶✮ ❛♥❞ ✇❥,❦ ❛r❡ t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✷ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❚❤❡ ❍❛❛r✬s s❝❛❧❡ ❛♥❞ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♠❛② ❜❡ r❡❝✉rs✐✈❡❧② ❝♦♠♣✉t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ s②♥t❤❡s✐s ❡q✉❛t✐♦♥s✱ ✉❥−✶,✷❦ = ✷−✶/✷(✉❥,❦ + ✇❥,❦) ✭✺✺✮ ✉❥−✶,✷❦+✶ = ✷−✶/✷(✉❥,❦ − ✇❥,❦). ✭✺✻✮ ❙♦✱ str✐❝t❧② ♣♦s✐t✐✈❡ s✐❣♥❛❧s ♠❛② ❜❡ ♠♦❞❡❧❡❞ ✐❢ ✉❥,❦ ≥ ✵ ❛♥❞ |✇❥,❦| ≤ ✉❥,❦ . ✭✺✼✮ ■t ✐s ♣♦ss✐❜❧❡ t♦ ❝❤♦♦s❡ ❛ st❛t✐st✐❝❛❧ ♠♦❞❡❧ ❢♦r ✇❥,❦ t❤❛t ✐♥❝♦r♣♦r❛t❡s t❤❡ ❝♦♥❞✐t✐♦♥ ✭✺✼✮✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✸ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❚❤❡ ▼❲▼ s♣❡❝✐✜❡s ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♠♦❞❡❧✱ ✇❥,❦ = ▼❥,❦✉❥,❦ , ✭✺✽✮ ✐♥ ✇❤✐❝❤ t❤❡ ♠✉❧t✐♣❧✐❡r ▼❥,❦ ♠❛② ❜❡ ♠♦❞❡❧❡❞ ❛s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ s②♠♠❡tr✐❝ β ❞✐str✐❜✉t✐♦♥ ✇✐t❤ s❤❛♣❡ ♣❛r❛♠❡t❡r ♣❥✱ ✐✳ ❡✳✱ ▼❥ ∼ β(♣❥, ♣❥)✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ▼❲▼ ✐s ❦♥♦✇♥ ❛s β✲▼❲▼ ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ▼❥,❦✬s ❛r❡ ♠✉t✉❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ ✉❥,❦ ✹✳

✹❘✐❡❞✐ ❡t ❛❧ ❤❛✈❡ ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞ ♦t❤❡r ❞✐str✐❜✉t✐♦♥s ❢♦r t❤❡ ♠✉❧t✐♣❧✐❡rs✳ ❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✹ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ ▼❥ ✐s ❣✐✈❡♥ ❜② ❱❛r[▼❥] = ✶ ✷♣❥ + ✶ . ✭✺✾✮ ■♥ t❤✐s ✇❛②✱ ❡q✉❛t✐♦♥s ✭✺✺✮ ❛♥❞ ✭✺✻✮ ♠❛② ❜❡ r❡✇r✐tt❡♥ ❛s ✉❥−✶,✷❦ = ✶ + ▼❥,❦ √ ✷

• ✉❥,❦

✭✻✵✮ ✉❥−✶,✷❦+✶ = ✶ − ▼❥,❦ √ ✷

• ✉❥,❦.

✭✻✶✮ ❚❤❡s❡ ❡q✉❛t✐♦♥s s❤♦✇ t❤❛t t❤❡ ▼❲▼ ✐s✱ ✐♥ ❢❛❝t✱ ❛ ❜✐♥♦♠✐❛❧ ❝❛s❝❛❞❡✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✺ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❚❤❡ ▼❲▼ ♠❛② ❛♣♣r♦①✐♠❛t❡ t❤❡ ❙❉❋ ♦❢ ❛ tr❛✐♥✐♥❣ s❡q✉❡♥❝❡ ❜② ♠♦❞❡❧✐♥❣ t❤❡ ✈❛r✐❛♥❝❡ ❞❡❝❛② ♦❢ t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts η❥ = ❱❛r[✇❥,❦] ❱❛r[✇❥−✶,❦] = ✷♣(❥−✶) + ✶ ♣(❥) + ✶ , ✭✻✷✮ t❤❛t ❧❡❛❞s t♦ ♣(❥−✶) = η❥ ✷ (♣(❥) + ✶) − ✶/✷ ✭✻✸✮ ❛♥❞ ♣(❥) = ✷♣(❥−✶) + ✶ η❥ − ✶ . ✭✻✹✮

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✻ ✴ ✻✼

▲♦♥❣ ▼❡♠♦r② ❚❡❧❡tr❛✣❝ ▼♦❞❡❧✐♥❣ ▼♦❞❡❧♦ ▼❲▼

❚❤❡ ▼❲▼ ❛ss✉♠❡s t❤❛t ✉❏−✶,✵ ✭t❤❡ ✏r♦♦t✑ s❝❛❧❡ ❝♦❡✣❝✐❡♥t✮ ✐s ❛♣♣r♦①✐♠❛t❡❧② ●❛✉ss✐❛♥✳ ❲❡ ❝❛♥ s❤♦✇ t❤❛t ♣(❥) ❝♦♥✈❡r❣❡s t♦ ♣−∞ = ❧✐♠

❥→−∞ ♣(❥) = ✷α − ✶

✷ − ✷α . ✭✻✺✮

❚❛❜❧❡✿ ❆s②♠♣t♦t✐❝ ✈❛❧✉❡s ♦❢ t❤❡ s❤❛♣❡ ♣❛r❛♠❡t❡r ♣ ♦❢ t❤❡ β ♠✉❧t✐♣❧✐❡rs ❛s ❢✉♥❝t✐♦♥ ♦❢ α ✭♦r ❍✮✳

α ✵.✶ ✵.✷ ✵.✺ ✵.✽ ♣ ✵.✵✼✼ ✵.✶✼✺ ✵.✼✵✼ ✷.✽✻ ❍ ✵.✺✺ ✵.✻ ✵.✼✺ ✵.✾ ❚❤❡ ▼❲▼ ♠♦❞❡❧ ❤❛s ♠✉❧t✐❢r❛❝t❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ✐s ❧♦❣♥♦r♠❛❧✳

❆♠❛③♦♥❛s ✭❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✮ ❚❡❧❡tr❛✣❝ ♠♦❞❡❧✐♥❣ ❛♥❞ ❡st✐♠❛t✐♦♥ ❙ã♦ P❛✉❧♦ ✶✶✴✷✵✵✽ ✻✼ ✴ ✻✼