第3章(2)

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Unformatted text preview: * 3 · ™ 3 ` (* ª · ªh j * @ ™ j 3.3 · ( ª · ` * ª ˜7 3.3.1 FT h • FT h Ù Ð * ª· ‡ 1è l … x(t ) = … ∞ ∑ x (t − kT ) k =− ∞ 0 x(t )h x0 (t )8úy·ª lim x(t ) = x0 (t ) T →∞ [email protected] 1 * 3 · ™ 3 ` (* ª · ªh j *[email protected] ª£ ·… FS h T = 4 ω0 = *ª@ ·£ … h π 2 x ( t) x(t ) = ∞ ak e jkω 0t ∑ k =− ∞ 2T1 1 a0 = = T 2 ak = sin( kω 0T1 ) sin( kπ 2) = kπ kπ [email protected] 2 * 3 · ™ 3 ` (* ª · ªh j T →* ∞… ª ¡¨ · 2π (1)ω 0 = →0 → T 1 1 T2 − jkω 0 t (2)ak = ∫ x(t )e dt = ∫ x0 (t )e − jkω 0t dt → 0 TT T −T 2 3 h x0(t) h FT h T2 lim Tak = lim ∫−T 2 x0 (t )e T →∞ T →∞ − jkω 0t ∞ dt = ∫ x0 (t )e − jω t dt = X 0 ( jω ) −∞ ∞ x(t ) ←→ X ( jω ) = ∫ x(t )e − jω t dt FT −∞ [email protected] ˜ FT 3 * 3 · ™ 3 ` (* ª · ªh j * 4 ¸÷ ª ·y IFT h ∞ x(t ) = ∑ x0 (t − kT ) = k =− ∞ 1 ak = T ∫ T2 −T 2 x(t )e ∞ − jkω 0 t X 0 ( jω ) = ∫ x0 (t )e −∞ ∞ ak e jkω 0t ∑ k =− ∞ 1 dt = T − jω t ∫ T2 dt = ∫ T2 −T 2 −T 2 1 ak = X 0 ( jkω 0 ) T 1∞ 1 jkω 0 t x(t ) = ∑ X 0 ( jkω 0 )e = T k = −∞ 2π x0 (t )e − jkω 0t dt x0 (t )e − jω t dt ∞ X 0 ( jkω 0 )e jkω 0tω 0 ∑ k =− ∞ [email protected] 4 * 3 · ™ 3 ` (* ª · ªh j ∞ 1 x0 (t ) = lim x(t ) = lim ∑ X 0 ( jkω 0 )e jkω 0tω 0 T →∞ 2π ω 0 →0 k = −∞ 1∞ = X 0 ( jω )e jω t dω 2π ∫−∞ 1∞ x(t ) = X ( jω )e jω t dω à ∫−∞ 2π IFTT 5 h FS h FT h (1) x(t ) = ∞ ak e jkω 0t ∑ k =− ∞ FS @Û E Ø Ú [email protected] 5 * 3 · ™ 3 ` (* ª · ªh j e jkωª 0tØ * ·ò y aª k Ø ·ò y 1 (2) x0 (t ) = 2π ∫ ∞ −∞ X 0 ( jω )e jω t dω FT C T C T e jω t X 0 ( jω )T e* jωò tØy ª· (3) x N (t ) = N ak e jkω* 0t òª Øy · ∑ x(t )h FS k =− N 1 ˆ x0 (t ) = 2π ∫ W −W X 0 ( j ω ) e j ω t dω [email protected] x0 * (tª )¬ · òy IFT 6 * 3 · ™ 3 ` (* ª · ªh j FT h IFT h FT ∞ x(t ) ←→ X ( jω ) = ∫ x(t )e − jω t dt FT −∞ 1 IFTT X ( jω ) ← → x(t ) = 2π IFT X (ª j· ωy )˜ò ∫ ∞ −∞ X ( j ω ) e j ω t dω x(t ) T X* (ª j· ωy ò)˜ ω š° X ( jω ) = X ( jω ) e jθ (ω ) X ( jω ) x(t ) θ (ω ) [email protected] x(t ) 7 * 3 · ™ 3 ` (* ª · ªh j 3.3.2 ª FT 8 · yù h ∫ ª· yù 8 1T x(t ) * ª·8 ùy 3.3.3· y ù 8 ª * (ª1)8y ù· x(t ) dt < ∞ ª · y (8 *ù X ( jω ) x(t ) 3 −∞ x(tª ù )8y *· 2 ∞ X ( jωª )8 · ùy @¨ T x(t ) = e t u (t ) ) X ( jω ) ª (8 · ùy ) ( ) x(t ) = u (t ) x(t ) = e jω 0t ω· 8 ª ùy FT x(t ) = e − at u (t )T a > 0 [email protected] 8 * 3 · ™ 3 ` (* ª · ªh j ∞ X ( jω ) = ∫ x(t )e − jω t −∞ ∞ =∫ e e T X ( jω ) = dt = ∫ e − at u (t )e − jω t dt −∞ − at − jω t 0 ∞ ∞ dt = ∫ e − ( a + jω ) t 0 − ( a + jω ) t 0 e = dt a + jω ∞ 1 = a + jω 1 a2 + ω 2 ω Arg[ X ( jω )] = − arctan a * (2y)d ª· õ x(t ) = e −a t T a>0 [email protected] 9 * 3 · ™ 3 ` (* ª · ªh j ∞ X ( jω ) = ∫ x(t )e − jω t −∞ ∞ =∫ e ∞ dt = ∫ e −∞ − ( a + jω ) t −a t e − j ω t dt 0 dt + ∫ e ( a − jω ) t dt −∞ 0 1 1 2a = + =2 a + jω a − j ω a +ω2 δ (t ) (3) * ª ùy · ∞ X ( jω ) = ∫ x(t )e −∞ ª(4)èù ·y − jω t ∞ dt = ∫ δ (t )e − jω t dt = 1 E x(t ) = 0 −∞ t <τ 2 other [email protected] 10 * 3 · ™ 3 ` (* ª · ªh j ∞ X ( jω ) = ∫ x(t )e − jω t −∞ − jω t τ 2 Ee = − jω = G• (5)÷ ª· yò −τ dt = ∫ τ2 −τ 2 Ee − jω t dt e jω τ 2 − e − j ω τ =E jω 2 2 2 jE sin(ω τ 2) = jω 2 E sin(ω τ 2) sin(ω τ 2) ω τ = Eτ = Eτ Sa ω ωτ 2 2 FT ←→ ¸y * ª ò· sin(ω c t ) x(t ) = πt [email protected] 11 * 3 · ™ 3 ` (* ª · ªh j * ª (÷ ·y X 1 ( jω ) 1 x1 (t ) = 2π 1 = 2π = ∞ ∫ X 1 ( j ω ) e jω t d ω −∞ ωc ∫ω − jω t e j 2π t e jω t dω c ωc sin ω c t = πt −ω c 1 ω < ω c sin(ω c t ) FT x(t ) = ←→ X ( jω ) = πt 0 other PË FT ←·→ ( ª y ÷* [email protected] 12 * 3 · ™ 3 ` (* ª · ªh j x(t ) = 1 (6ª )È÷ ·y X ( jωª )È÷ * ·y x(t ) = 1 G à T X 1 ( jω ) = δ (ω ) 1 x1 (t ) = 2π ∫ ∞ −∞ δ (ω )e jω t 1 dω = 2π FT x(t ) = 1 ←→ X ( jω ) = 2πδ (ω ) x(t ) = e jω 0t (* 7ª )∪ · y T X 1 ( jω ) = δ (ω − ω 0 ) 1 x1 (t ) = 2π e jω 0 t δ (ω − ω 0 )e jω t dω = ∫−∞ 2π ∞ [email protected] 13 * 3 · ™ 3 ` (* ª · ªh j FT x(t ) = e jω 0t ←→ X ( jω ) = 2πδ (ω − ω 0 ) T FT cos(ω 0t ) ←→ π [δ (ω − ω 0 ) + δ (ω + ω 0 )] FT sin (ω 0t ) ←→ − jπ [δ (ω − ω 0 ) − δ (ω + ω 0 )] 3.4 X x(t ) FT Ø x(t ) = T ∞ ak e jkω 0t ∑ k =− ∞ 2π ω0 = T x(t ) ←→ FT ∞ ∑ 2π a δ (ω − kω ) k =− ∞ k [email protected] 0 14 * 3 · ™ 3 ` (* ª · ªh j 3− 2 ª8 ·… ¢ FT x(t ) = T = 4 ω0 = π 2 x(t ) = ∑a e k =− ∞ k jkω 0t ak e jkω 0t ∑ k =− ∞ sin( kω 0 T1 ) sin( kπ 2) ak = = kπ kπ ∞ ∞ ( k ≠ 0) 1 a0 = 2 kπ ←→ 2π ∑ ak δ ω − 2 k =− ∞ FT ∞ [email protected] 15 * 3 · ™ 3 ` (* ª · ªh j 3 − 4ùy·ª* FT δ T (t ) = [email protected] ∞ ∑ δ (t − kT ) k =− ∞ 16 * 3 · ™ 3 ` (* ª · ªh j δ T (t ) T ω 0 = 2π T x(t ) = ∞ ak e jkω 0t ∑ k =− ∞ 1 ak = T ∫ T2 −T 2 2π FT δ T (t ) ←→ T δ T (t )e − jkω 0 t 1 dt = T ∞ ∑ δ (ω − kω k =− ∞ 0 ∫ T2 −T 2 ) = ω0 δ (t )e − jkω 0t 1 dt = T ∞ ∑ δ (ω − kω k =− ∞ [email protected] 0 ) = ω 0δ ω 0 (ω ) 17 * 3 · ™ 3 ` (* ª · ªh j *ªx ·ó y P122 h 3.1(2) P122 h 3.2(1)(2)(5) P123 h 3.5(1)(4)(5) P123 h 3.6(1) (2) [email protected] 18 ...
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