第3章(3)

第3ç«&n...

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Unformatted text preview: * sª1 c3 P °* ª · ·³ ∞ X ( jω ) = ∫ x(t )e − jω t dt −∞ ∞ X ( jω ) ω =0 = X ( j 0) = ∫ x(t )e −∞ − j0 t ∞ dt = ∫ x(t )dt −∞ ∞ X ( jω ) ω =π = X ( jπ ) = ∫ x(t )e − jπ t dt −∞ ∫ ∞ ∫ ∞ −∞ −∞ x(t )e x(t )e −j ω t 2 jω t ω dt = X j 2 dt = X ( − jω ) ∞ x(t )e − j 2ω t dt = X ( j 2ω ) ∫−∞ ∞ x(t )e − j (ω −ω 0 ) t dt = X ( j (ω − ω 0 ) ) ∫−∞ [email protected] 1 * sª1 c3 P °* ª · ·³ 3.5 FT s 3.5.1 FT FT x1 (t ) ←→ X 1 ( jω ) x2 (t ) ←→ X 2 ( jω ) FT ax1 (t ) + bx2 (t ) ←→ aX 1 ( jω ) + bX 2 ( jω ) 3.5.2 FT x(t − t0 ) ←→ X ( jω )e − jω t0 FT x(t ) ←→ X ( jω ) ∫ ∞ −∞ x(t − t0 )e − jω t ∞ dt = ∫ x(τ )e −∞ − jω (τ + t 0 ) dτ = e [email protected] − jω t 0 ∫ ∞ −∞ x(τ )e − jω τ dτ 2 * sª1 c3 P °* ª · ·³ t 3 − 5t x(t ) = cos(ω 0t + θ )t X ( jω ) 1 x(t ) = cos(ω 0t + θ ) = cosθ cos(ω 0t ) − sin θ sin(ω 0t ) 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 )] X ( jω ) = π cosθ [δ (ω − ω 0 ) + δ (ω + ω 0 )] + jπ sin θ [δ (ω − ω 0 ) − δ (ω + ω 0 )] = π [e jθ δ (ω − ω 0 ) + e − jθ δ (ω + ω 0 )] [email protected] 3 * sª1 c3 P °* ª · ·³ θ 2 t x(t ) = cos(ω 0t + θ ) = cos ω 0 t + ω 0 t FT cos( ω 0t ) ←→ π [δ (ω − ω 0 ) + δ (ω + ω 0 )] X ( jω ) = e jω θ ω0 π [δ (ω − ω 0 ) + δ (ω + ω 0 )] = π [e jθ δ (ω − ω 0 ) + e − jθ δ (ω + ω 0 )] 3.5.3 FT t x(t ) ←→ X ( jω )t FT x(t )e jω 0 t ←→ X ( j (ω − ω 0 )) [email protected] 4 * sª1 c3 P °* ª · ·³ ∫ ∞ −∞ t t x(t )e jω 0 t − jω t e ∞ dt = ∫ x(t )e − j (ω −ω 0 )t dt = X ( j (ω − ω 0 )) −∞ 1 x(t )cos(ω 0t ) ←→ [ X ( j (ω − ω 0 )) + X ( j (ω + ω 0 ))] 2 1 FT x(t )sin (ω 0t ) ←→ [ X ( j (ω − ω 0 )) − X ( j (ω + ω 0 ))] 2j FT x(t ) = sin(10π t ) cos(50π t )t πt sin(10π t ) x0 (t ) = πt t X ( jω ) 1 ω < 10π X 0 ( jω ) = other 0 [email protected] 5 * sª1 c3 P °* ª · ·³ 1 x(t ) = x0 (t ) cos(50π t ) ←→ [ X 0 ( j (ω − 50π )) + X 0 ( j (ω + 50π ))] 2 FT 3.5.4ª ¶ X Ê* FT x* (t ) ←→ X * (− jω ) ª ¶ 1Ξ Êo * ∫ ∞ −∞ * x (t )e − jω t dt = ∫ ∞ −∞ [ x(t )e ] jω t * dt = ∫ x(t )e −∞ [email protected] ∞ jω t * dt = X * (− jω ) 6 * sª1 c3 P °* ª · ·³ * ª 2h Ê ¶ X ( jω ) = X * ( − j ω ) x(t ) = x ∗ (t ) x(t ) ∈ R X ( jω ) = Re[ X ( jω )] + j Im[ X ( jω )] X ( jω ) X * (− jω ) = Re[ X (− jω )] − j Im[ X (− jω )] Re[ X ( jω )] = Re[ X (− jω )] 1 Œ¸ x(t ) Im[ X ( jω )] = − Im[ X (− jω )] * ª¶η 6 Ê Œ¸ ª¶η 6 Ê ω ω¶ η ª6 Ê [email protected] 7 * sª1 c3 P °* ª · ·³ 2 x(t ) £ ω£ ωÊ ªø ¶ ∞ X ( jω ) = ∫ x(t )e − jω t −∞ dt = ∫ x(t )[ cos(ω t ) − jsin(ω t )] dt ∞ −∞ ∞ ∞ −∞ −∞ = ∫ x(t )cos(ω t )dt − j ∫ x(t )sin(ω t )dt 3 ª¶ Ê t x(t ) £ * ªω ¶t Ê t 4t x(t ) £ X ( j* ω¶ ) ªt Ê ª ¶ t øÊ * ªω øÊ ¶t ω ωt X ( j* ω¶ t)øÊ ª [email protected] 8 * sª1 c3 P °* ª · ·³ x(t ) ∈ R 5 FT xe (t ) ←→ Re[ X ( jω )] x(t ) = xe (t ) + xo (t ) FT xo (t ) ←→ j Im[ X ( jω )] x(t ) = e − at u (t )t a > 0 t 1 −a t xe (t ) = e 2 1 a − jω x(t ) ←→ X ( jω ) = =2 a + jω a +ω2 FT [email protected] a xe (t ) ←→ 2 a +ω2 FT 9 * sª1 c3 P °* ª · ·³ 3.5.5 ° dx(t ) FT ←→ jωX ( jω ) dt * ª 1♦ ¶k Ê 1 x(t ) = 2π ∫ ∞ −∞ X ( jω ) e jω t dω dx(t ) 1 = dt 2π ∫ ∞ −∞ j ω X ( j ω ) e j ω t dω d k x(t ) FT k ←→( jω ) X ( jω ) dt k * ª 2♦ ¶ Êz ∫ t −∞ FT x(τ )dτ ←→ [email protected] X ( jω ) + π X ( j 0)δ (ω ) jω 10 * sª1 c3 P °* ª · ·³ * ªÊ ¶ 8 FT x1 (t ) t x(t ) = ∫ x1 (τ )dτ −∞ ωτ X 1 ( jω ) = Eτ Sa = 2 Sa (ω ) 2 X 1 ( jω ) 2 Sa (ω ) X ( jω ) = + π X 1 ( j 0)δ (ω ) = + 2π δ (ω ) jω jω [email protected] 11 * sª1 c3 P °* ª · ·³ dx(t ) x1 (t ) = dt X 1 ( jω ) = jωX ( jω ) = 2 Sa (ω ) + jω ⋅ 2πδ (ω ) = 2 Sa (ω ) 3− 6 t u (t ) FT x1 (t ) = δ (t )t X 1 ( jω ) = 1 X 1 ( jω ) 1 + π X 1 ( j 0)δ (ω ) = + π δ (ω ) u (t ) = ∫ δ (τ )dτ ←→ −∞ jω jω t t 3 − 8t FT sgn(t ) t FTt [email protected] 12 * sª1 c3 P °* ª · ·³ t 2 2 + 2πδ (ω ) − 2πδ (ω ) = sgn(t ) = 2u (t ) − 1 ←→ jω jω FT 1 x(t ) = lim T → ∞ 2T dx1 (t ) dx2 (t ) = = δ (t ) dt dt 1 3 X 1 ( jω ) = + 2π δ (ω ) ⋅ jω 2 ∫ T −T x(t )dt j ω X 1 ( j ω ) = j ωX 2 ( j ω ) = 1 1 1 X 2 ( jω ) = + 2π δ (ω ) ⋅ jω 2 [email protected] 13 * sª1 c3 P °* ª · ·³ 1 1 + π δ (ω ) + 2π δ (ω ) = + 3π δ (ω ) x1 (t ) = u (t ) + 1 ←→ jω jω FT 3.5.6ª ¶ ø *Ê s x(t ) ←→ X ( jω )s FT 1 ω x(at ) ←→ X j a a FT ω −j τ 1 ω 1∞ FT − jω t a dτ = X j x(at ) ←→ ∫ x(at )e dt = ∫ x(τ )e −∞ a a a −∞ ∞ ª ¶ Ê o a = −1 FT x(−t ) ←→ X (− jω ) [email protected] 14 * sª1 c3 P °* ª · ·³ 3.5.7 FT t x(t ) ←→ X ( jω )t FT X ( jt ) ←→ 2π x(−ω ) ∞ X ( jω ) = ∫ x(t )e − jω t dt −∞ ∞ X ( jτ ) = ∫ x(t )e − jτ t −∞ ∞ X ( jt ) = ∫ x(ω )e −∞ 1 = 2π ∫ ∞ −∞ ∞ dt = ∫ x(ω )e − jτ ωdω −∞ − jω t ∞ dω = ∫ x(−ω )e jω t dω −∞ 2π x(−ω )e jω t dω [email protected] 15 * sª1 c3 P °* ª · ·³ 2 t 3 − 10t x(t ) = 2 t t +1 x1 (t ) = e −t t X ( jω ) X 1 ( jω ) = 2 ω 2 +1 2 FT ←→ 2π x1 (−ω ) = 2π e − −ω = 2π e − ω X 1 ( jt ) = 2 t +1 3.5.8ª ¶ H *Ê dX ( jω ) − jtx (t ) ←→ dω x(t ) ←→ X ( jω ) FT ∞ X ( jω ) = ∫ x(t )e −∞ − jω t FT dt ∞ dX ( jω ) = ∫ [ − jtx(t )] e − jω t dt −∞ dω [email protected] 16 * sª1 c3 P °* ª · ·³ X ( jω ) t 3 − 11t x(t ) = te − at u (t )t − at x1 (t ) = e u (t ) s 1 X 1 ( jω ) = jω + a dX 1 ( jω ) j − jtx1 (t ) = − jte u (t ) ←→ =− dω ( jω + a ) 2 − at FT 1 te u (t ) ←→ ( jω + a ) 2 − at 2 t e u (t ) ←→ ( jω + a ) 3 2 − at FT FT n! t e u (t ) ←→ ( jω + a ) ( n +1) n − at FT [email protected] 17 * sª1 c3 P °* ª · ·³ * ªÊ ¶ È P123 s 3.7 P123 s 3.8(2)(5)(10) P124 s 3.9(1)(2) P124 s 3.10(1)(3) [email protected] 18 ...
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This note was uploaded on 08/22/2011 for the course EE 201 taught by Professor Yhm during the Spring '05 term at Zhejiang University.

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