❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❇⑩❈❍ ❑❍❖❆ ❍⑨ ◆❐■ ◆●❯❨➍◆ ❆◆❍ ✣⑨■ ❚➑❈❍ ❈❍❾P ❙❯❨ ❘❐◆● ❋❖❯❘■❊❘ ❈❖❙■◆❊✱ ❋❖❯❘■❊❘ ❙■◆❊ ❚❍❮■ ●■❆◆ ❘❮■ ❘❸❈ ❱⑨ Ù◆● ❉Ö◆● ◆❣➔♥❤✿ ❚♦→♥ ❤å❝ ▼➣ sè✿ ✾✹✻✵✶✵✶ ❚➶▼ ❚➁❚ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✷✵ ❈æ♥❣ tr ữủ t t ì ữớ ữợ P ❚❍❷❖ P❤↔♥ ❜✐➺♥ ✶✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ P❤↔♥ ❜✐➺♥ ✷✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ P❤↔♥ ữủ trữợ ỗ t s rữớ t ỗ ❣✐í✱ ♥❣➔② ✳✳✳✳✳ t❤→♥❣ ✳✳✳✳✳ ♥➠♠ ✷✵✷✵✳ ❈â t❤➸ t➻♠ ❤✐➸✉ ❧✉➟♥ →♥ t↕✐ t❤÷ ✈✐➺♥✿ ✶✳ ❚❤÷ ✈✐➺♥ ❚↕ ◗✉❛♥❣ ❇û✉ ✲ ❚r÷í♥❣ ✣❍❇❑ ❍➔ ◆ë✐ ✷✳ ❚❤÷ ✈✐➺♥ ◗✉è❝ ❣✐❛ ❱✐➺t ◆❛♠ ❉❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝æ♥❣ ❜è ❝õ❛ ▲✉➟♥ →♥ ✶✳ ◆✳ ❆✳ ❉❛✐ ❛♥❞ ◆✳ ❳✳ ❚❤❛♦ ✭✷✵✶✽✮✱ ●❡♥❡r❛❧✐③❡❞ ❝♦♥✈♦❧✉t✐♦♥s ✇✐t❤ ✇❡✐❣❤t✲ ❢✉♥❝t✐♦♥ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ ❋♦✉r✐❡r ❝♦s✐♥❡ ❛♥❞ s✐♥❡ tr❛♥s❢♦r♠s✱ ❆♥♥❛❧❡s ❯♥✐✈✳ ❙❝✐✳ ❇✉❞❛♣❡st✳✱ ❙❡❝t✳ ❈♦♠♣✳✱ ❱♦❧✳ ✹✼✱ ♣♣✳ ✷✷✼✲✷✸✼✳ ✷✳ ◆✳❳✳ ❚❤❛♦✱ ❱✳❑✳ ❚✉❛♥ ❛♥❞ ◆✳❆✳ ❉❛✐ ✭✷✵✶✽✮✱ ❉✐s❝r❡t❡✲t✐♠❡ ❋♦✉r✐❡r ❝♦✲ s✐♥❡ ❝♦♥✈♦❧✉t✐♦♥✱ ■♥t✳ ❚r❛♥s✳ ✫ ❙♣❡❝✳ ❋✉♥❝t✳ ✭❙❈■❊✮✱ ❱♦❧✳✷✾✭✶✶✮✱ ♣♣✳ ✽✻✻✲✽✼✹✳ ✸✳ ◆✳❳✳ ❚❤❛♦ ❛♥❞ ◆✳❆✳ ❉❛✐ ✭✷✵✶✽✮✱ ❉✐s❝r❡t❡✲t✐♠❡ ❋♦✉r✐❡r s✐♥❡ ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠✱ ❏✳ ▼❛t❤✳ ❆♣♣❧✳✱ ❱♦❧✳✶✻✭✷✮✱ ♣♣✳ ✺✶✲✻✷✳ ✹✳ ◆✳❳✳❚❤❛♦✱ ❱✳❑✳❚✉❛♥ ❛♥❞ ◆✳❆✳ ❉❛✐ ✭✷✵✷✵✮✱ ❆ ❞✐s❝r❡t❡ ❝♦♥✈♦❧✉t✐♦♥ ✐♥✲ ✈♦❧✈✐♥❣ ❋♦✉r✐❡r s✐♥❡ ❛♥❞ ❝♦s✐♥❡ s❡r✐❡s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s✱ ■♥t✳ ❚r❛♥s✳ ✫ ❙♣❡❝✳ ❋✉♥❝t✳ ✭❙❈■❊✮✱ ❱♦❧✳ ✸✶✭✸✮✱ ♣♣✳ ✷✹✸✲✷✺✷✳ ▼Ð ✣❺❯ ✶✳ ▲à❝❤ sû ✈➜♥ ✤➲ ✈➔ ❧➼ ❞♦ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ❆✳ P❤➨♣ ❜✐➳♥ ✤ê✐ t➼❝❤ ♣❤➙♥ ✤è✐ ✈ỵ✐ ❧ỵ♣ ❤➔♠ ❦❤↔ t➼❝❤ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝â ❞↕♥❣ ∞ (F f )(x) = F [f ](x) = √ 2π e−ixy f (y)dy, f ∈ L1 (R) ✭✵✳✶✮ −∞ ◆➳✉ g(x) = (F f )(x) ∈ L1(R) t❛ ❝â ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷đ❝ ♥❤÷ s❛✉ ∞ f (x) = (F −1 g)(x) = F −1 [g](x) = √ 2π eixy g(y)dy ✭✵✳✸✮ −∞ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ f ∈ L1(R+) ❧➔ ❤➔♠ ❝❤➤♥ ❤♦➦❝ ❤➔♠ ❧➫✱ t❛ ♥❤➟♥ ✤÷đ❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ ❝â ❞↕♥❣ ∞ (Fc f )(y) = Fc [f ](y) = π f (x) cos(xy)dx, f ∈ L1 (R+ ), ✭✵✳✺✮ f (x) sin(xy)dx, f ∈ L1 (R+ ) ✭✵✳✻✮ ∞ (Fs f )(y) = Fs [f ](y) = π ❚➼❝❤ ❝❤➟♣ ✈➔ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❚➼❝❤ ❝❤➟♣ ✤➛✉ t✐➯♥ ✤÷đ❝ ①➙② ❞ü♥❣ ❧➔ t➼❝❤ ❝❤➟♣ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ f ✈➔ g ✤è✐ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝â ❞↕♥❣ ∞ (f ∗ g)(x) = √ F 2π f (y)g(x − y)dy, x ∈ R ✭✵✳✾✮ −∞ ◆➠♠ ✶✾✺✶✱ ■✳ ◆✳ ❙♥❡❞❞♦♥ ①➙② ❞ü♥❣ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ ❝❤➤♥ f ✈➔ g ✤è✐ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ ❝â ❞↕♥❣ ∞ (f ∗ g)(x) = √ Fc 2π f (y)[g(x + y) + g(x − y)]dy, x ∈ R+ , ✶ ✭✵✳✶✷✮ ∞ (f ∗ g)(x) = √ Fs 2π f (u)[g(|x − u|) − g(x + u)]du, x > ✭✵✳✶✺✮ P❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ tê♥❣ q✉→t ✤è✐ ✈ỵ✐ ❧ỵ♣ ❤➔♠ ❦❤↔ t➼❝❤ ❝â ❞↕♥❣ ∞ [k1 (x + y) + k2 (x − y)] f (y)dy = g(x), x > 0, f (x) + ✭✵✳✶✾✮ tr♦♥❣ ✤â g, k1, k2 ❧➔ ♥❤ú♥❣ ❤➔♠ ✤➣ ❜✐➳t✱ f ❧➔ ➞♥ ❤➔♠✳ ❱ỵ✐ ❝ỉ♥❣ ❝ư t➼❝❤ ❝❤➟♣✱ ♠ët sè ợ ữỡ tr t t tr trữớ ủ t õ t ữủ ữợ ❞↕♥❣ ✤â♥❣✳ ❇✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈ỵ✐ ❧ỵ♣ ❤➔♠ ❦❤↔ tê♥❣ ❜ð✐ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ rớ r ố ợ t x(n) ữủ ①→❝ ✤à♥❤ ∞ X(e ) ≡ FDT {x(n)} = iω x(n)e−iωn , x(n) ∈ C, ∀n, ✭✵✳✷✵✮ X(eiω )eiωn dω, ω ∈ [−π, π], ✭✵✳✷✶✮ −∞ ✈➔ ❝â ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝ ♥❤÷ s❛✉ π −1 x(n) ≡ FDT {X(eiω )} = 2π −π tr♦♥❣ ✤â ω ❧➔ ❜✐➳♥ t❤ü❝✱ ❝→❝ t➼♥ ❤✐➺✉ ✤➛✉ ✈➔♦ x(n) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ω✳ ❚➼❝❤ ❝❤➟♣ ✈ỵ✐ ❧ỵ♣ ❤➔♠ ❦❤↔ tê♥❣ ❚➼❝❤ ❝❤➟♣ ✤è✐ ✈ỵ✐ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ❤❛✐ ❞➣② x(n) ✈➔ y(n) ❝â ❞↕♥❣ ♥❤÷ s❛✉ ∞ (x ∗ y)(n) = x(m)y(n − m), −∞ < n < ∞, ✭✵✳✷✷✮ m=−∞ t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ FDT {x(n) ∗ y(n)}(eiω ) = FDT {x(n)}(eiω ) · FDT {y(n)}(eiω ), ✭✵✳✷✸✮ ✈ỵ✐ x(n), y(n) ∈ C, −∞ < n < ∞✱ ✈➔ ✤➥♥❣ t❤ù❝ P❛rs❡✈❛❧ π ∞ x(n)y(n) = 2π n=−∞ X(eiω )Y ∗ (eiω )dω, −π ✷ ✭✵✳✷✹✮ ✈ỵ✐ y∗(n) ❧➔ ❧✐➯♥ ❤đ♣ ♣❤ù❝ ❝õ❛ y(n) ✈➔ ❝â Y ∗(e−iω ) = FDT {y(n)} tợ ữ õ ổ tr ✈➲ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ♥â✐ ❝❤✉♥❣ ✈➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ♥â✐ r✐➯♥❣ ❝ơ♥❣ ♥❤÷ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ t❤í✐ rớ r ụ ữ ữủ tợ r ỡ s õ t tr ữợ ự ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ❝❤♦ ▲✉➟♥ →♥ ✈ỵ✐ t➯♥ ❣å✐ ✧❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ rí✐ r↕❝ ✈➔ ù♥❣ ❞ư♥❣ ✧✳ ✷✳ ▼ư❝ ✤➼❝❤✱ ✤è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ▼ö❝ ✤➼❝❤✿ ◆❣❤✐➯♥ ❝ù✉ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈ỵ✐ ❝→❝ ❞➣② t➼♥ ❤✐➺✉ ❜❛♥ ✤➛✉ ❧➔ ❝→❝ ❞➣② ❝❤➤♥ ❧➫ ✤è✐ ①ù♥❣✳ ❚ø ✤â ①➙② ❞ü♥❣ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✤è✐ ✈ỵ✐ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ t➼❝❤ ❝❤➟♣ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ✣→♥❤ ❣✐→ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ t❤í✐ ❣✐❛♥ rí✐ r ự ởt số ợ ữỡ tr t rớ r ã ố tữủ s rở t➼❝❤ ❝❤➟♣✱ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✤è✐ ✈ỵ✐ ❝→❝ ❜✐➳♥ ✤ê✐ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❝♦s✐♥❡✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r ữỡ tr t rớ r ã P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ▲➔ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝❀ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝❀ ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r ã Pữỡ ự r sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ♣❤÷ì♥❣ ♣❤→♣ t♦→♥ tû✱ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ t➼❝❤ ♣❤➙♥✳ ❇➯♥ ❝↕♥❤ ✤â✱ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝ơ♥❣ ✤÷đ❝ sû ❞ư♥❣✳ ✹✳ ❈➜✉ tró❝ ✈➔ ❝→❝ ❦➳t q✉↔ ❝õ❛ ▲✉➟♥ →♥ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ▲✉➟♥ →♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜❛ ❝❤÷ì♥❣ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➛♥ ❞ị♥❣ tr♦♥❣ ▲✉➟♥ →♥✳ ❈ư t❤➸ ❧➔ ❝→❝ ❞➣② t➼♥ ❤✐➺✉ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ ❤➺ t❤è♥❣ t➼♥ ❤✐➺✉✱ ❤➺ t❤è♥❣ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳t t❤í✐ ❣✐❛♥✱ t➼❝❤ ✸ ❝❤➟♣ ❝õ❛ ❤➺ t❤è♥❣ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳♥ t❤í✐ ❣✐❛♥✱ ❜✐➳♥ rr tớ rớ r ỳ ỵ ♠➺♥❤ ✤➲ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ▲✉➟♥ →♥✳ ❈❤÷ì♥❣ ✷✿ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ①➙② ❞ü♥❣ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈➔ t➼❝❤ ❝❤➟♣ ✤è✐ ✈ỵ✐ ❜✐➳♥ ✤ê✐ ✤â✳ ◆❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t t♦→♥ tû ❝õ❛ ❝→❝ t s rở ữ sỹ tỗ t t ❜à ❝❤➦♥✱ ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛✱ ✤➥♥❣ t❤ù❝ P❛rs❡✈❛❧✳ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈ỵ✐ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❧✐➯♥ q✉❛♥ tỵ✐ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ tr➯♥ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❞➣②✳ ◆❤➟♥ ✤÷đ❝ ✤à♥❤ ỵ rớ r t tự rí✐ r↕❝ ✈➔ ❝→❝ ✤→♥❤ ❣✐→ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤✉➞♥ ✤è✐ ✈ỵ✐ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ ✤â✳ ❈❤÷ì♥❣ ✸✿ P❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè ♣❤➨♣ ❜✐➳♥ ✤ê✐ t➼❝❤ ♣❤➙♥ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ ❧✐➯♥ q✉❛♥ tỵ✐ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ①➙② ❞ü♥❣ ✤÷đ❝ tr♦♥❣ ❈❤÷ì♥❣ ✷✱ ♥❤➟♥ ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✤â ❧➔ ✉♥✐t❛✳ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝ t➼❝❤ ❝❤➟♣✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✤➣ ①➙② ❞ü♥❣ ✤÷đ❝ ✈➔♦ ✈✐➺❝ ❣✐↔✐ ✈➔ ✤→♥❤ ❣✐→ ♥❣❤✐➺♠ ❝õ❛ ♠ët ✈➔✐ ợ ữỡ tr ữỡ tr t rớ r ị t q t ữủ tr →♥ ❳➙② ❞ü♥❣ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈ỵ✐ ❤➔♠ trå♥❣ ✈➔ t➼❝❤ ❝❤➟♣ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ tø ✤â ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ◆❣❤✐➯♥ ❝ù✉ ✈➔ t❤✐➳t ❧➟♣ ✤÷đ❝ ♥❤ú♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤✉➞♥ ✤è✐ ✈ỵ✐ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ t❤í✐ ❣✐❛♥ rớ r ợ ỹ ữủ ỵ rớ r ỵ tr ứ õ ✤÷đ❝ ù♥❣ ❞ư♥❣ ❣✐↔✐ ✈➔ ✤→♥❤ ❣✐→ ♥❣❤✐➺♠ ❝õ❛ ♠ët số ợ ữỡ tr t rớ r õ ú t ỵ tt tớ ❣✐❛♥ rí✐ r↕❝✱ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ▲✉➟♥ →♥ ❞ü❛ tr➯♥ ✹ ❝æ♥❣ tr➻♥❤ ✤➣ ❝æ♥❣ ❜è✱ tr♦♥❣ ✤â ❝â ✷ ❜➔✐ ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ t❤✉ë❝ ❞❛♥❤ ♠ö❝ ■❙■✱ ✶ ❜➔✐ ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ q✉è❝ t➳ ✈➔ ✶ ❜➔✐ t❤✉ë❝ t↕♣ ❝❤➼ q✉è❝ ❣✐❛✳ ✹ ❈❤÷ì♥❣ ✶ ❚✃◆● ◗❯❆◆ ❱➋ ❇■➌◆ ✣✃■ ❋❖❯❘■❊❘ ❚❍❮■ ●■❆◆ ❘❮■ ❘❸❈ ✶✳✶ ❚➼♥ ❤✐➺✉ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ ❤➺ t❤è♥❣ ✶✳✶✳✶ ❚➼♥ ❤✐➺✉ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❚➼♥ ❤✐➺✉ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ t ữợ số ởt số x✱ tr♦♥❣ ✤â sè t❤ù n ❝õ❛ ❞➣② ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ x(n)✱ ✤÷đ❝ ✈✐➳t ð ❞↕♥❣ ♥❤÷ s❛✉ x = {x(n)}, −∞ < n < ∞, ✭✶✳✶✮ ð ✤â n ❧➔ ♠ët sè ♥❣✉②➯♥✳ ❉➣② ✤ì♥ ✈à ♠➝✉ ❧➔ ❞➣② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ 0, ♥➳✉ n = ✭✶✳✷✮ δ(n) = 1, ♥➳✉ n = ❚ê♥❣ q✉→t ❤ì♥✱ ❜➜t ❦ý ❝❤✉é✐ ♥➔♦ ✤➲✉ ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ ữợ x(k)(n k) x(n) = k=−∞ ✶✳✶✳✷ ❈→❝ ❤➺ t❤è♥❣ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳♥ ▼ët ❤➺ t❤è♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✤÷đ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ♠ët t♦→♥ tû ❚ ❧➔ ♥❤✐➺♠ ✈ö ❜✐➳♥ ✤ê✐ ❞➣② ✈➔♦ x(n) t❤➔♥❤ ❞➣② r❛ y(n)✳ ❈❤ó♥❣ t❛ ❝â t❤➸ sỷ ỵ t sỷ s T {x(n)} = y(n), ✭✶✳✼✮ ❤♦➦❝ T x(n) −→ y(n) ❑❤✐ ❤➺ t❤è♥❣ ❧➔ ❤➺ t❤è♥❣ t✉②➳♥ t➼♥❤ ✈➔ ❜➜t ❜✐➳♥✱ t❛ ❝â q✉❛♥ ❤➺ s❛✉✿ T [δ(n)] = h(n), ✺ T [δ(n − k)] = h(n − k) = hk (n), ∞ ∞ ⇒ y(n) = x(k)h(n − k) x(k)hk (n) = ✭✶✳✶✶✮ k=−∞ k=−∞ ✣→♣ ù♥❣ ①✉♥❣ h(n) s➩ ✤➦❝ tr÷♥❣ ❤♦➔♥ t♦➔♥ ❝❤♦ ♠ët ❤➺ t❤è♥❣ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳♥ ✈➔ t❛ ❝â q✉❛♥ ❤➺ s❛✉ ∞ ✈ỵ✐ ♠å✐ n ∈ Z, x(k)h(n − k) = x(n) ∗ h(n), y(n) = ✭✶✳✶✷✮ k=−∞ ✤÷đ❝ ❣å✐ ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ x(n) ✈➔ h(n)✳ ✶✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rớ r ữủ ợ t tr t ✸✼✱ ✸✽✱ ✸✾❪✱ ❝â ❞↕♥❣ ∞ X(ω) ≡ FDT {x(n)}(ω) = x(n)e−iωn , ✭✶✳✶✹✮ n=−∞ ✈➔ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝ ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rí✐ r↕❝ π −1 x(n) = FDT {X(ω)} = 2π X(ω)eiωn dω ✭✶✳✶✺✮ −π ð ✤➙② X(ω) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π✱ ❞➣② t➼♥ ❤✐➺✉ ✤➛✉ ✈➔♦ x(n) ❝â t❤➸ ❧➔ t❤ü❝ ự tổ tữớ t t tợ t ❤✐➺✉ ♣❤ù❝✮✳ ✶✳✷✳✷ ❚➼♥❤ ❝❤➜t ❝õ❛ ❜✐➳♥ ✤ê✐ ❚➼❝❤ ❝❤➟♣ ❚➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ❤❛✐ ❞➣② t➼♥ ❤✐➺✉ x(n) ✈➔ y(n) ❝â ❞↕♥❣ ∞ x(n) ∗ y(n) = x(m)y(n − m), ∀n ∈ Z, ✭✶✳✷✷✮ m=−∞ ✻ ✈➔ ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ G(ω) = FDT {x(n) ∗ y(n)}(ω) = X(ω)Y (ω) ✭✶✳✷✸✮ ❱✐ ♣❤➙♥ tr♦♥❣ ♠✐➲♥ t➛♥ sè F DT nx(n) ←→ i dX(ω) d tự Prss ỵ |x(n)|2 = n=−∞ 2π |X(ω)|2 dω, ✭✶✳✷✺✮ 2π tr♦♥❣ ✤â |X(ω)|2 ✤÷đ❝ ❣å✐ ❧➔ ♠➟t ✤ë ♥➠♥❣ ❧÷đ♥❣ ❝õ❛ t➼♥ x(n) ỵ r ỵ ✶✳✷✳✶✳ ✐✮ ❈❤♦ x(n) ∈ ✈➔ Φ(z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ ♠✐➲♥ t➛♥ sè ❝õ❛ FDT {x(n)}(ω)✳ ❑❤✐ ✤â✱ Φ FDT {x(n)}(ω) ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ❞➣② y(n) ∈ 1(N0) ♥➔♦ ✤â ✈➔ ❧➟♣ t❤➔♥❤ ❝❤✉é✐ ❤ë✐ tö t✉②➺t ✤è✐✳ ✐✐✮ ✣➦❝ ❜✐➺t✱ ♥➳✉ FDT {x(n)}(ω) = ✈ỵ✐ ♠é✐ ω ∈ [0, π] t tỗ t số z(n) (Z) s❛♦ ❝❤♦ FDT {z(n)}(ω) = ✳ F {x(n)}(ω) (Z) DT ỵ x(n) tở ❦❤æ♥❣ ❣✐❛♥ x ∗ y ∞ FsDT ✈➔ y(n) ∈ ✱ ❤ì♥ ♥ú❛ o (N0 ) ✳ ❚❤➻ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ (N0 ) o ∞ (N0 ) ≤2 x lim (x ∗ y)(n) = · y 2, FsDT n→∞ ✭✷✳✶✹✮ ◆❣♦➔✐ r❛✱ t❛ ❝â ✤➥♥❣ t❤ù❝ P❛rs❡✈❛❧ s❛✉ ✤➙② π (x ∗ y)(n) = FsDT π Xs (ω)Yc (ω) sin(nω)dω, n ≥ ✭✷✳✶✺✮ ỵ ỵ rớ r ❈❤♦ p, q, r > 1✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x(n) ∈ ✱ y(n) ∈ ✱ h(n) ∈ o p (N0 ) q (N0 ) r (N0 ) ✈➔ p1 + 1q + 1r = 2✱ t❤➻ · y · h r ∞ (x ∗ y)(n) · h(n) ≤ x FsDT n=0 p q ✭✷✳✶✾✮ ❍➺ q✉↔ ✷✳✷✳✶ ✭❇➜t ✤➥♥❣ t❤ù❝ ❦✐➸✉ ❨♦✉♥❣ rí✐ r↕❝✮✳ ✣➦t p, q, r > 1✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (x ∗ y)(n) ∈ FsDT ✳ ◆➳✉ ✱ ❤ì♥ ♥ú❛ t❛ ❝â 1 p + q o r (N0 ) = 1+ r x y FsDT r ỵ sỷ r➡♥❣ x(n) ∈ (x ∗ y)(n) ∈ FsDT x(n) ∈ ≤ x p ✱ o p (N0 ) y(n) ∈ · y q ✱ t❤➻ q (N0 ) ✭✷✳✷✸✮ ✱ y(n) ∈ 1(N0)✳ ❑❤✐ ✤â ✱ ✈➔ ❝â ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ s❛✉ o (N0 ) o (N0 ) FsDT {(x ∗ y)(n)}(ω) = 2FsDT {x(n)}(ω) · FcDT {y(n)}(ω), ω ∈ [0, π] FsDT ✭✷✳✷✹✮ ❍ì♥ ♥ú❛✱ x ∗ y FsDT ≤2 x · y ✭✷✳✷✺✮ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ x(n) ✈➔ y(n) ❧➔ ❝→❝ ❞➣② ❦❤ỉ♥❣ ➙♠ ✭❤♦➦❝ ❦❤ỉ♥❣ ❞÷ì♥❣✮✳ ✶✵ ✷✳✷✳✷ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈ỵ✐ ❤➔♠ trå♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✷✳ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈ỵ✐ ❤➔♠ trå♥❣ γ(ω) = sin(ω) ❝õ❛ ❝→❝ ❞➣② x(n), y(n) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ∞ γ (x ∗ y)(n) = x(m)[y(|m + n − 1|) FsDT ✭✷✳✸✸✮ m=1 +y(|n − m − 1|) − y(n + m + 1) − y(|n − m + 1|), ♥➳✉ ❝❤✉é✐ ❤ë✐ tư ✈ỵ✐ n ỵ x(n), y(n) ∈ γ trå♥❣ (x F∗ y)(n) ∈ o1(N0) ✈➔ ✱ ❦❤✐ ✤â t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈ỵ✐ ❤➔♠ o (N0 ) sDT ✭✷✳✸✹✮ γ ||x ∗ y||1 ≤ 4||x||1 · ||y||1 FsDT ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ❝→❝ ❞➣② t➼♥ ❤✐➺✉ x(n) ✈➔ y(n) ❧➔ ❝→❝ ❞➣② ❦❤æ♥❣ ➙♠ ✭❤♦➦❝ ❦❤ỉ♥❣ ❞÷ì♥❣✮✳ ❍ì♥ ♥ú❛✱ t❛ ❝â ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ γ FsDT (x ∗ y)(ω) = sin ω · FcDT {x(n)}(ω) · FcDT {y(n)}(ω), ω ∈ [0, π] FsDT ❍➺ q✉↔ ✷✳✷✳✷✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② t➼♥ ❤✐➺✉ t❤✉ë❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈ỵ✐ ❤➔♠ trå♥❣ γ(ω) = sin(ω) ✤è✐ ✈ỵ✐ ❜✐➳♥ ✤ê✐ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r s✐♥❡ ❧➔ ❣✐❛♦ ❤♦→♥ γ γ (x ∗ y)(n) = (y ∗ x)(n) ∀ω ∈ [0, π] ✭✷✳✹✼✮ F F sDT ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✸✳ ❑❤æ♥❣ ❣✐❛♥ n ∈ N0 ✱ t❤ä❛ ♠➣♥ o (N0 ) sDT n (N0 , e ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè x(n) ✈ỵ✐ ∞ n (N0 , e |x(n)en | < ∞ ) = x = {x(n)} : ✭✷✳✹✽✮ n=0 ❑❤æ♥❣ ❣✐❛♥ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ 1(N0, en) x(0) = ỵ ỵ ❚✐t❝❤♠❛r❝❤✮ ❈❤♦ γ❝→❝ ✤→♣ ù♥❣ ①✉♥❣ x(n), y(n) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t t❤✉ë❝ o1(N0, en)✳ ❑❤✐ ✤â (x F ∗ y)(n) ≡ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x(n) ≡ ❤♦➦❝ y(n) ≡ ✈ỵ✐ ♠å✐ n ≥ 0✳ ✶✶ o n (N0 , e ) sDT ✷✳✸ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✷✳✸✳✶ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ❤❛✐ ❞➣② x(n) ✈➔ y(n) ❧➔ ♠ët ❞➣② (x F∗ cDT y)(n)✱ ①→❝ ✤à♥❤ ❜ð✐ ∞ x ∗ y (n) = x(k)[y(n + k) + y(|k − n|)sign(k − n)], n ≥ 0, FcDT ✭✷✳✺✵✮ k=1 ♥➳✉ ❝❤✉é✐ ❤ë✐ tư ✈ỵ✐ ♠å✐ n ≥ 0✳ ỵ x(n), y(n) o (N0 ) ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ t❤➻ (x F∗ y)(n) ∈ o (N0 ) cDT FcDT (x ∗ y)(n) (ω) = 2FsDT {x(n)}(ω) · FsDT {y(n)}(ω) FcDT ✈➔ t❛ õ ỵ ỵ trs ❤❛✐ ❞➣② ❣✐↔♠ t❤❡♦ ❝➜♣ sè ♠ô x(n), y(n) ∈ o1(N0)✳ ❑❤✐ ✤â (x F∗ ♥➳✉ x(n) ≡ 0✱ ❤♦➦❝ y(n) ≡ 0✳ cDT y)(n) ≡ ∀ n ∈ N0 ỵ ỵ ❦✐➸✉ ❨♦✉♥❣ rí✐ r↕❝✮✳ ❈❤♦ p, q, r > 1✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x(n) ∈ ✱ y(n) ∈ o p (N0 ) ✱h∈ o q (N0 ) o r (N0 ) ✈➔ p1 + 1q + 1r = 2✱ t❤➻ ∞ (x ∗ y)(n) · h(n) ≤ x n=0 FcDT p · y q · h r ✭✷✳✻✵✮ ❍➺ q✉↔ ✷✳✸✳✶ ✭❇➜t ✤➥♥❣ t❤ù❝ ❦✐➸✉ ❨♦✉♥❣ rí✐ r↕❝✮✳ ✣➦t p, q, r > 1✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ + 1q = + 1r ✳ ◆➳✉ x(n) ∈ (x ∗ y)(n) ∈ r (N0 )✱ ❤ì♥ ♥ú❛ t❛ ❝â p ✱ o p (N0 ) y(n) ∈ ✱ t❤➻ o q (N0 ) FcDT x ∗ y FcDT r ≤ x ✶✷ p · y q ✭✷✳✻✹✮ ✷✳✸✳✷ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈ỵ✐ ❤➔♠ trå♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✷✳ ❚➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❝❤♦ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈ỵ✐ ❤➔♠ trå♥❣ γ(ω) = sin ω ❝õ❛ ❤❛✐ ❞➣② sè x(n) ✈➔ y(n) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ∞ γ (x ∗ y)(n) = FcDT x(m)[y(n + m − 1) + y(|n − m + 1|) ✭✷✳✻✺✮ m=1 − y(n + m + 1) − y(|n − m − 1|)], ♥➳✉ ❝❤✉é✐ ❤ë✐ tư ✈ỵ✐ n ỵ x(n) (N0 ) ✈➔ o (N0 ) ✈➔ y(n) ∈ ✱ t❤➻ (N0 ) γ (x ∗ y)(n) ∈ FcDT ✭✷✳✻✻✮ γ ||x ∗ y||1 ≤ 4||x||1 · ||y||1 FcDT ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ❞➣② t➼♥ ❤✐➺✉ x(n) ✈➔ y(n) ❧➔ ❝→❝ ❞➣② ❦❤æ♥❣ ➙♠ ✭❤♦➦❝ ❦❤ỉ♥❣ ❞÷ì♥❣✮✳ ❍ì♥ ♥ú❛✱ t❛ ❝â ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ γ FcDT {(x ∗ y)(n)}(ω) = sin ω · FsDT {x(n)}(ω) · FcDT {y(n)}(ω), FcDT ω ∈ [0, ] ỵ ỵ tr x(n), y(n) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t t❤✉ë❝ 1(N0, en)✳ ❑❤✐ ✤â (x F ∗ ✈ỵ✐ ♠å✐ n ≥ 0✳ y)(n) ≡ cDT ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x(n) ≡ ❤♦➦❝ y(n) ≡ ✷✳✹ ❚➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✳ ❚➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ❤❛✐ ❞➣② x(n) ✈➔ y(n) ❧➔ ♠ët ❞➣② (x F∗♦ cDT ♦ ①→❝ ✤à♥❤ ❜ð✐ ∞ (x ∗ y)(n) = FcDT y)(n)✱ x(m)[y(n + m) + y(|n − m|)] + x(0)y(n), m=1 ♥➳✉ ❝❤✉é✐ ❤ë✐ tư ✈ỵ✐ ♠å✐ n ≥ 0✳ ✶✸ n ≥ 0, ỵ x(n), y(n) (N0 ) t❤➻ (x F∗♦ y)(n) ∈ (N0 ) cDT ✈➔ ✭✷✳✽✺✮ ♦ ||x ∗ y||1 ≤ 2||x||1 · ||y||1 FcDT ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ x(n) ✈➔ y(n) ❧➔ ❝→❝ ❞➣② ❦❤ỉ♥❣ ➙♠ ✭❤♦➦❝ ❦❤ỉ♥❣ ❞÷ì♥❣✮✳ ❍ì♥ ♥ú❛✱ t❛ ❝â ✤➥♥❣ t❤ù❝ ♥❤➙♥ tû ❤â❛ s❛✉✿ ♦ FcDT {(x ∗ y)(n)}(ω) = 2FcDT {(x)(n)}(ω) · FcDT {(y)(n)}(ω) F cDT ỵ x(n), y(n) ✳ ❚❤➻ (N0 ) ♦ x ∗ y FcDT ∞ ≤2 x ✭✷✳✽✾✮ · y ✣➥♥❣ t❤ù❝ ✤↕t ✤÷đ❝ ❦❤✐ ❝→❝ ✈❡❝tì x(n) ✈➔ y(n) ❧➔ ♣❤ư t❤✉ë❝ t✉②➳♥ t➼♥❤✳ ❍ì♥ ♥ú❛✱ t❛ ❝â ✤➥♥❣ t❤ù❝ P❛rs❡✈❛❧ π (x ∗ y)(n) = FcDT π ♦ Xc ()Yc () cos(n)d, n ỵ ỵ trs x(n), y(n) ❣✐↔♠ t❤❡♦ ❝➜♣ sè ♠ô✳ ❑❤✐ ✤â (x F∗ y)(n) ≡ cDT ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x(n) ≡ ❤♦➦❝ y(n) ≡ 0✳ ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✷ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ✤➣ ✤↕t ✤÷đ❝✿ ❼ ❳➙② ❞ü♥❣ ❤❛✐ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ ❤❛✐ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈ỵ✐ ❤➔♠ trå♥❣ ✈ỵ✐ ❤❛✐ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ ♠ët t➼❝❤ ❝❤➟♣ ✈ỵ✐ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ❈❤ù♥❣ ♠✐♥❤ sü tỗ t t tự tỷ ❤â❛✱ ✤➥♥❣ t❤ù❝ P❛rs❡✈❛❧ ❝õ❛ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ ♥➔② tr➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ p(N0) ✈➔ o p (N0 ) ✈ỵ✐ p = 1, 2✳ ❼ ❈❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤✉➞♥ ❝õ❛ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ tr➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ p(N0) ✈➔ op(N0), p = 1, ỵ tr rớ r↕❝ ❝❤♦ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣❀ t➼❝❤ ❝❤➟♣ t❤í✐ rớ r ỵ rớ r t ✤➥♥❣ t❤ù❝ ❦✐➸✉ ❨♦✉♥❣ rí✐ r↕❝ ✈ỵ✐ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✳ ✶✹ ❈❤÷ì♥❣ ✸ P❍➆P ❇■➌◆ ✣✃■ ❑■➎❯ ❚➑❈❍ ❈❍❾P ❙❯❨ ❘❐◆● ❚❍❮■ ●■❆◆ ❘❮■ ❘❸❈ ❱⑨ P❍×❒◆● ❚❘➐◆❍ ❚❖❊P▲■❚❩✲❍❆◆❑❊▲ ❘❮■ ❘❸❈ ✸✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡✱ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✸✳✶✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❳➨t ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✤è✐ ✈ỵ✐ ❝→❝ ❞➣② t➼♥ ❤✐➺✉ ✤➛✉ ✈➔♦ x(n) ✈➔ t➼♥ ❤✐➺✉ ✤➛✉ r❛ y(n) ❝â ❞↕♥❣ ♥❤÷ s❛✉ TsDT : x →x ∗ k FsDT ∞ y(n) =TsDT {x(n)} = x(m) k(|n − m|) − k(n + m) , n ≥ ✭✸✳✶✮ m=1 ỵ ỵ ts rớ r ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ TsDT tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✸✳✶✮ ❧➔ ✉♥✐t❛ tr♦♥❣ o (N0 ) ❧➔ |FcDT {k(n)}(ω)| = ω ∈ [0, π] ✭✸✳✷✮ ❍ì♥ ♥ú❛✱ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝ ❝â ❞↕♥❣ −1 x(n) = TsDT {y(n)} = (y ∗ k)(n) FsDT ∞ y(m) k(|n − m|) − k(n + m) , n ≥ 0, = m=1 ð ✤➙② k ❧➔ ❧✐➯♥ ❤ñ♣ ♣❤ù❝ ❝õ❛ k✳ ✶✺ ✭✸✳✸✮ ❱➼ ❞ư ✸✳✶✳✶✳ ❱ỵ✐ k(n) = FcDT {k(n)}(ω) = − 2i e ω ✈➔ ✭✸✳✶✮ ❧➔ ✉♥✐t❛ tr♦♥❣ o2(N0)✳ i i(−1)n − , ✈ỵ✐ (4n2 − 1)π |FcDT {k(n)}(ω)| = 21 ✳ ❉♦ ✤â✱ ♠å✐ n ≥ 0✳ ❚❤➻ ❜✐➳♥ ✤ê✐ t➼❝❤ ❝❤➟♣ ✸✳✶✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❳➨t ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✤è✐ ✈ỵ✐ ❞➣② t➼♥ ❤✐➺✉ ✤➛✉ ✈➔♦ x(n) ✈➔ t➼♥ ❤✐➺✉ ✤➛✉ r❛ y(n) ❞↕♥❣ TcDT : x → k ∗ x, FcDT ∞ y(n) = TcDT {x(n)} = x(m) k(m + n) + k(|m − n|)s✐❣♥(m − n) , ∀n ∈ N0 , ✭✸✳✻✮ m=1 ð ✤➙② ❞➣② k(n) ✤÷đ❝ ❣å✐ ❧➔ ♥❤➙♥ ❝õ❛ tớ rớ r ỵ ỵ ts rớ r ✤➸ ❜✐➳♥ ✤ê✐ TcDT tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✸✳✻✮ ❧➔ ✉♥✐t❛ tr♦♥❣ o2(N0) ❧➔ |FsDT {k(n)}(ω)| = , ∀ω ∈ [0, π] ✭✸✳✼✮ ❍ì♥ ♥ú❛✱ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝ ❝â ❞↕♥❣ −1 x(n) = TcDT {y(n)} = (k ∗ y)(n) FcDT ∞ y(|n − m|) − y(n + m) k(m), n ≥ 0, = ✭✸✳✽✮ m=1 tr♦♥❣ ✤â k ❧➔ ❧✐➯♥ ❤đ♣ ♣❤ù❝ ❝õ❛ k✳ ❱➼ ❞ư ✸✳✶✳✷✳ ❚❛ ❝ơ♥❣ ❝â kn = (4n22n− 1)π (−1)n + i , n ≥ 0✳ FsDT {k(n)}(ω) = 2i eiω ✈➔ |FsDT {k(n)}(ω)| = ✭✸✳✻✮ ❧➔ ✉♥✐t❛ tr♦♥❣ o2(N0)✳ ✶✻ ❚❤➻ ✳ ❉♦ ✤â✱ ❜✐➳♥ ✤ê✐ t➼❝❤ ❝❤➟♣ ✸✳✶✳✸ P❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❚❛ ①➨t ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✤è✐ ✈ỵ✐ ❞➣② t➼♥ ❤✐➺✉ ✤➛✉ ✈➔♦ x(n) ✈➔ ❞➣② t➼♥ ❤✐➺✉ ✤➛✉ r❛ y(n) ❞↕♥❣ ♥❤÷ s❛✉ o ♦ T cDT :x(n) → (k ∗ x)(n), FcDT ∞ o y(n) =T cDT {x(n)} = x(0)k(n) + x(m) k(n + m) + k(|n − m|) , n ≥ 0, ✭✸✳✶✶✮ m=1 ð ✤➙②✱ ❞➣② k(n) ✤÷đ❝ ❣å✐ ❧➔ ♥❤➙♥ ❝õ❛ ❜✐➳♥ ✤ê✐ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ✣à♥❤o ỵ ỵ ts rớ r ❝➛♥ ✈➔ ✤õ ✤➸ ❜✐➳♥ ✤ê✐ T cDT tr♦♥❣ ✭✸✳✶✶✮ ❧➔ ✉✐♥t❛ tr♦♥❣ l2(N0) ❧➔ |Kc (ω)| ≡ |FcDT {k(n)}(ω)| = , ω ∈ [0, π] ✭✸✳✶✷✮ ❍ì♥ ♥ú❛✱ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝ ❝â ❞↕♥❣ o ♦ x(n) = T −1 cDT {y(n)} = (k ∗ y)(n) FcDT ∞ y(m)[k(n + m) + k(|n − m|)] + y(0)k(n), n ≥ 0, = ✭✸✳✶✸✮ m=1 tr♦♥❣ ✤â k ❧➔ ❧✐➯♥ ❤đ♣ ♣❤ù❝ ❝õ❛ k✳ ❱➼ ❞ư ✸✳✶✳✸✳ ❚÷ì♥❣ tü ♥❤÷ ✈➼ ❞ư ✸✳✶✳✶✱ ✈ỵ✐ kn , n ≥ 0✳ i(−1)n − (4n − 1)π |FcDT {k(n)}(ω)| = 21 ✳ ❉♦ ✤â✱ = ❚❤➻ FcDT {k(n)}(ω) = − 2i e ω ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✭✸✳✶✶✮ ❧➔ ✉♥✐t❛ tr♦♥❣ 2(N0)✳ i ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ❚ø ữỡ tr t ố ợ ợ t s❛✉ ❦❤✐ rí✐ r↕❝ ❤â❛ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ❝â ❞↕♥❣ ✿ ∞ [k1 (n + m) + k2 (n − m)]y(m) = h(n), n ∈ N0 x(n) + m=0 ✶✼ ✭✸✳✹✮ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tổ ởt số ợ ữỡ tr tr ợ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ♥❤➙♥ ✈➔ ✈➳ ♣❤↔✐✳ ✸✳✷✳✶ Pữỡ tr t rớ r ợ t t ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ tê♥❣ q✉→t ✭✸✳✶✻✮✱ tr♦♥❣ ✤â ♥❤➙♥ k1 (n) ✈➔ k2 (n) ❧➔ trò♥❣ ♥❤❛✉ k1 (n) = k2 (n) = k(n)✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ ∞ x(n) + ✭✸✳✶✼✮ [k(n + m) + k(n − m)]y(m) = h(n), n ∈ N0 m=0 Ð ✤â k(n), h(n) ∈ 1(N0) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t ✈➔ x(n) ∈ 1(N0) ❧➔ ❞➣② ❝➛♥ t➻♠✳ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ♥➔②✱ t❛ ①➨t ❇ê ✤➲ s❛✉ ❇ê ỵ r rr ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✮ ❈❤♦ x(n) ∈ 1(N0) ✈➔ Φ(z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ ♠✐➲♥ t➛♥ sè ❝õ❛ FcDT {x(n)}(ω)✳ ❑❤✐ ✤â✱ Φ FcDT {x(n)}(ω) ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ❞➣② y(n) ∈ 1(N0) ♥➔♦ ✤â ✈➔ ❧➟♣ t❤➔♥❤ ♠ët ❝❤✉é✐ ❤ë✐ tö t✉②➺t ✤è✐✳ ✣➦❝ ❜✐➺t✱ ♥➳✉ FcDT {x(n)}(ω) = ✈ỵ✐ ộ [0, ] tỗ t z(n) 1(N0) s❛♦ ❝❤♦ FcDT {z(n)}(ω) = FcDT {x(n)}(ω) ❛✮ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ ∞ x(m)[k(n + m) + k(|n − m|)] + x(0)k(n) = z(n), n ≥ 0, ✭✸✳✶✽✮ m=1 ∞ x(m)[k(n + m) + k(|n − m|)] + x(0)k(n) = z(n), x(n) + m=1 n ≥ 0, ✭✸✳✶✾✮ ð ✤â y(n), z(n) ❧➔ ❝→❝ ❞➣② ❣✐→ trà ♣❤ù❝ ✤➣ ❜✐➳t✱ ✈➔ x(n) ❧➔ ❞➣② ❝➛♥ t➻♠✳ ✶✽ ✣à♥❤ ỵ k(n), z(n) FcDT {k(n)}() = ✈ỵ✐ ♠é✐ ω ∈ [0, π]✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ ❞↕♥❣ ✭✸✳✶✽✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t x(n) ∈ 1(N0) ♦ x(n) = (u ∗ z)(n), ✭✸✳✷✵✮ F (N0 ) cDT ð ✤â u(n) ∈ (N0 ) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ π u(n) = π cos(nω) dω 2FcDT {k(n)}(ω) ❱➼ ❞ö ✸✳✷✳✶✳ t ữỡ tr t rớ r ợ ❞➣② k(n), z(n) ∈ ❝â ❞↕♥❣ (N0 ) k(n) = z(n) = ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â eπ (−1)n − , n ≥ 0, π(1 + n2 ) −1 + (−1)n π2 , n ≥ ✈➔ z(0) = πn2 ♥❣❤✐➺♠ ❞✉② ♥❤➜t x(n) ∈ 1(N0) ð ❞↕♥❣ + n2 − e−π + π + n2 (π − 1) (−1)n x(n) = , n 2(1 + n2 )2 ỵ k(n), z(n) ∈ ✱ ✈➔ + 2Kc(ω) = ợ ộ [0, ] õ ữỡ tr ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ✭✸✳✶✾✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ t❤✉ë❝ 1(N0) tr♦♥❣ ❞↕♥❣ (N0 ) ♦ x(n) = z(n) − (v ∗ z)(n), FcDT ð ✤➙② v(n) ∈ (N0 ) ✭✸✳✷✷✮ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ π v(n) = π FcDT {k(n)}(ω) cos(nω)dω + 2FcDT {k(n)}(ω) ❱➼ ❞ư ✸✳✷✳✷✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ❞↕♥❣ ✭✸✳✶✾✮✱ ✈ỵ✐ ❝→❝ ❞➣② k(n), z(n) ∈ (N0 ) ①→❝ ✤à♥❤ ❜ð✐ −1 + eπ (−1)n k(n) = , n≥0 π(1 + a2 ) ✶✾ −1 + (−1)n , n≥1 πn2 ✈➔ z(0) = π2 ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t t❤✉ë❝ 1(N0) ❝â ❞↕♥❣ z(n) = − n2 − + n2 (−1 + π) + π e−π (−1)n x(n) = , n ≥ π(1 + n2 )2 ❱➼ ❞ö ✸✳✷✳✸✳ ❳➨t ữỡ tr t rớ r ợ k(n), z(n) ∈ ❑❤✐ ✤â ♣❤÷ì♥❣ (N0 ) ①→❝ ✤à♥❤ ❜ð✐ + (−1)n π k(n) = , n ≥ ✈➔ z(0) = , πn2 8(−1)n 4π z(n) = , n ≥ ✈➔ z(0) = −π + n2 tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t t❤✉ë❝ 1(N0) ❝â ❞↕♥❣ −2 + 2(−1)n x(n) = ,n ≥ πn2 ✈➔ x0 = −π + π2 ❜✮ ❚❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③ ✲ ❍❛♥❦❡❧ rí✐ r↕❝ ❝â ❞↕♥❣ ♥❤÷ s❛✉✿ ∞ x(m)[k(|n − m|) − k(n + m)] = h(n), x(n) + n ≥ ✭✸✳✷✾✮ m=1 Ð ✤â k(n), h(n) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t✱ x(n) t ỵ k(n) 1(N0), h(n) ∈ o1(N0)✱ ✈➔ 1+2FcDT {k(n)}(ω) = ✈ỵ✐ ♠é✐ ω ∈ [0, π]✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✾✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t x(n) ∈ o (N0 ) ❞↕♥❣ x(n) = h(n) − (h ∗ w)(n), ✭✸✳✷✻✮ F ð ✤â w(n) ∈ l1(N0) ①→❝ ✤à♥❤ ❜ð✐ sDT w(n) = π π FcDT {k(n)}(ω) cos(nω) dω + 2FcDT {k(n)}() Pữỡ tr t ợ ♣❤↔✐ ✤➦❝ ❜✐➺t ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ∞ x(n) + [k1 (n + m) + k2 (|n − m|)]x(m) = g(m), m=1 ✷✵ n ∈ N0 , ✭✸✳✸✺✮ ð ✤➙② ❝→❝ k1(n), k2(n) tũ ỵ tọ trữợ ỵ sû r➡♥❣ g2(n), k1(n), k2(n), ∈ 1(N0), g(n) = g1(n) + g2 (n) ✈➔ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✈➔ g1 (n) = ð ✤â l(n) ∈ ✭✸✳✹✵✮ π FcDT {k1 (n) + k2 (n)}(ω) = 0, ∀ω ∈ [0, π], 1+ ♦ π (g2 ∗ l) − g2 FcDT ✭✸✳✹✶✮ ∗ (k1 − k2 ) (n), FsDT ✱ ①→❝ ✤à♥❤ ❜ð✐ (N0 ) π 2π l(n) = FcDT {k1 (n) + k2 (n)}(ω) cos(nω)dω + π2 FcDT {k1 (n) + k2 (n)}(ω) ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ rí✐ r↕❝ ✭✸✳✸✺✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❞↕♥❣ x(n) = g2(n) − 12 (g2 F l)(n) (N0 ) ữợ cDT ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ❛✮ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ x(n) + ∞ ∞ u(m)[y(|n − m|) − y(n + m)] = z(n), m=1 x(m) v(n + m) + v(|m − n|)sign(m − n) + y(n) = h(n), ✭✸✳✺✵✮ m=1 ð ✤â u(n), v(n), z(n), h(n) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t✱ x(n), y(n) t ỵ u(n), v(n), h(n) ✈➔ z(n) ∈ o1(N0) t❤➻ (u F∗ v)(n) ∈ (N0 )✳ ❱➔ ♥➳✉ − 4FsDT {u(n)}(ω) · FsDT {v(n)}(ω) = ∀ ω ∈ [0, π]✱ ❦❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✺✵✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t t❤✉ë❝ o1(N0) ❞↕♥❣ cDT x(n) = z(n) + (z ∗ k)(n) − (u ∗ h)(n) − (u ∗ h) ∗ k (n) FsDT ♦ FsDT FcDT FcDT FsDT FsDT ♦ y(n) = h(n) + (h ∗ k)(n) − (v ∗ z)(n) − (v ∗ z) ∗ k (n), ✷✶ FcDT FcDT ✭✸✳✺✶✮ ð ✤â k(n) ∈ ✱ ①→❝ ✤à♥❤ ❜ð✐✿ (N0 ) π k(n) = π 2FsDT {u(n)}(ω) · FsDT {v(n)}(ω) cos(nω)dω − 4FsDT {u(n)}(ω) · FsDT {v(n)}(ω) ✭✸✳✺✷✮ ❜✮ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ ∞ x(n) + y(m)[u(n + m − 1) + u(|n − m + 1|) m=0 −u(|m + n + 1|) − u(|n − m − 1|)] = z(n) ∞ v(k)[x(|n − k|) − x(n + k)] + y(n) = w(n), ✭✸✳✺✽✮ k=0 ð ✤â ❝→❝ ❞➣② u(n), z(n), v(n) ✈➔ w(n) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t✱ x(n) y(n) t ỵ ✸✳✷✳✻✳ ◆➳✉ u(n), v(n), z(n), w(n) ∈ o1(N0) ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ − sin ω · FsDT {v(n)}(ω) · FcDT {u(n)}(ω) = 0, ∀ω ∈ [0, π] ✭✸✳✺✾✮ ❑❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✺✽✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t t❤✉ë❝ o1(N0) ð ❞↕♥❣ γ γ ♦ ♦ x(n) = z(n) − (w ∗ u)(n) + (h ∗ z)(n) + h ∗ (w ∗ u) (n), FcDT FcDT FcDT ♦ w(n) − (v ∗ z)(n) + (w ∗ h)(n) − (v ∗ z) ∗ h (n), y(n) = ✈ỵ✐ h(n) ∈ FcDT FsDT FsDT FsDT FcDT ✱ ①→❝ ✤à♥❤ ❜ð✐ ✭✸✳✻✵✮ (N0 ) π h(n) = π sin ω · FsDT {v(n)}(ω) · FcDT {u(n)}(ω) cos(nω)dω − sin ω · FsDT {v(n)}(ω) · FcDT {u(n)}(ω) ❝✮ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ∞ x(n) + u(m)[y(|n + m − 1|) + y(|n − m − 1|) m=0 −y(n + m + 1) − y(|n − m + 1|) = z(n) ∞ v(k)[x(k + n) + x(|k − n|)sign(k − n)] + y(n) = w(n), ✭✸✳✻✻✮ k=0 ð ✤â ❝→❝ ❞➣② u(n), z(n), v(n) ✈➔ w(n) ❧➔ ❝→❝ ❞➣② ✤➣ ❜✐➳t✱ x(n) ✈➔ y(n) t ỵ ◆➳✉ u(n), z(n), v(n), w(n) ∈ (N0 ) ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ − sin ω · FsDT {v(n)}(ω) · FcDT {u(n)}(ω) = 0, ∀ω ∈ [0, π] ❑❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✻✻✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t x(n) ✈➔ y(n) ∈ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ γ x(n) = z(n) + z ∗ h (n) − u ∗ w (n) − u ∗ w w(n) + w ∗ h (n) − v ∗ z (n) − v ∗ z FsDT ♦ y(n) = FcDT ❱ỵ✐✱ h(n) ∈ (N0 ) π h(n) = FsDT π FcDT ①→❝ ✤à♥❤ ❜ð✐ γ FsDT FcDT ✱ (N0 ) ∗ h (n) FsDT ♦ ∗ h (n) FcDT ✭✸✳✻✼✮ sin ω · FsDT {v(n)}(ω) · FcDT {u(n)}(ω) cos(nω)dω − sin ω · FsDT {v(n)}(ω) · FcDT {u(n)}(ω) ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✸ ❈→❝ t q t ữủ ã ỹ ❝➛♥ ✈➔ ✤õ ✤➸ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❧➔ ✉♥✐t❛ tr♦♥❣ o2(N0) ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❧➔ ✉♥✐t❛ tr♦♥❣ 2(N0) ✈➔ t❤✐➳t ❧➟♣ ❝ỉ♥❣ t❤ù❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝✳ ỗ tớ r ử t sỹ tỗ t ự ró ỡ sỹ tỗ t ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ • ●✐↔✐ r❛ ♥❣❤✐➺♠ ❜✐➸✉ ❞✐➵♥ q✉❛ ❝→❝ t➼❝❤ s rở t ợ ỹ ữủ ợ ữỡ tr t rớ r ợ t ữỡ tr t rớ r ợ t ❦ý ✈➔ ✈➳ ♣❤↔✐ ✤➦❝ ❜✐➺t✱ ✤÷❛ r❛ ♠ët sè ✈➼ ❞ư ❝ư t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ợ t ởt số ữỡ tr ữỡ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝ ♥❤÷ ❤➺ ✭✸✳✺✵✮✱ ✭✸✳✺✽✮ ✈➔ ❤➺ ✭✸✳✻✻✮ ❝ơ♥❣ ❣✐↔✐ r❛ ♥❣❤✐➺♠ t❤ỉ♥❣ q✉❛ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✤â✳ ✣→♥❤ ❣✐→ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ q✉❛ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈ỵ✐ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✈➔ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ rớ r ợ ỹ ữủ ❱⑨ ❑■➌◆ ◆●❍➚ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ →♥ ✤↕t ✤÷đ❝✿ ✶✳ ❳➙② ❞ü♥❣ ♠ët sè t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❝â trå♥❣ ✈➔ t➼❝❤ ❝❤➟♣ ✤è✐ ✈ỵ✐ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ❚ø ✤â✱ tr➯♥ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠ ①→❝ ✤à♥❤ t❛ ♥❤➟♥ ✤÷đ❝ ♠ët sè t➼♥❤ ❝❤➜t t♦→♥ tû✱ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤✉➞♥✱ ✤➥♥❣ t❤ù❝ ♥❤➙♥ tỷ õ tự Prs ỵ tr ỵ t tự ố ✈ỵ✐ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡✱ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ✷✳ ◆❤➟♥ ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✤è✐ ✈ỵ✐ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝♦s✐♥❡ ✈➔ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❧➔ ✉♥✐t❛ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ o2(N0)✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❧➔ ✉♥✐t❛ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ 2(N0)✱ t❤✐➳t ❧➟♣ ❝ỉ♥❣ t❤ù❝ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝✳ ❚ø ✤â ❝❤♦ ✈➼ sỹ tỗ t ợ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ s✉② rë♥❣✱ ❦✐➸✉ t➼❝❤ ❝❤➟♣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ♥➯✉ tr➯♥✳ ✸✳ ❚❤✐➳t ❧➟♣ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈ỵ✐ ❝→❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ❋♦✉r✐❡r ❝♦s✐♥❡✱ ❋♦✉r✐❡r s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❝â trå♥❣ ✈➔ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ 2(N0)✱ o2(N0)✳ ●✐↔✐ ✈➔ ✤→♥❤ ❣✐→ ♥❣❤✐➺♠ ❝õ❛ ♠ët sè ❧ỵ♣ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❚♦❡♣❧✐t③✲❍❛♥❦❡❧ rí✐ r↕❝✳ ✷✹ ... rớ r ỵ tr ứ õ ♥❤➟♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ ❣✐↔✐ ✈➔ ✤→♥❤ ❣✐→ ♥❣❤✐➺♠ ❝õ❛ ởt số ợ ữỡ tr t rớ r õ ú t ỵ tt t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣ s✉② rë♥❣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ❝❤➟♣✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤... o2(N0) ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❦✐➸✉ t➼❝❤ ❝❤➟♣ ❋♦✉r✐❡r ❝♦s✐♥❡ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ❧➔ ✉♥✐t❛ tr♦♥❣ 2(N0) ✈➔ tt ổ tự ữủ ỗ tớ ❝❤➾ r❛ ❝→❝ ✈➼ ❞ö ❝ö t❤➸ ♠✐♥❤ ❤å❛ ❝❤♦ sỹ tỗ t ự ró ỡ sỹ tỗ t ❜✐➳♥ ✤ê✐