电子科技大学电子工程学院：《信号与系统》课程教学资源（PPT课件讲稿，英文版）Chapter 4 The Continuous-Time Fourier Transform

Chapter 4 Fourier transform 94.3 Properties of the Fourier Transforms 54.3. 1 Linearity x(o F,XUjo) y(k< F>rja ax(t) +by(o F>axjo)+brljo 2 x,() 2() 101t-2 0 2 t F 4sino-2sin20 x(t)<1 16

16 Chapter 4 Fourier Transform §4.3 Properties of the Fourier Transforms §4.3.1 Linearity x(t)⎯→X(j) y(t)⎯→Y(j) F F ax(t)+ by(t)⎯→aX(j)+ bY(j) F -2 -1 0 1 2 t −1 1 x(t) -2 0 2 t 1 x (t) 2 x (t) 2 1 -1 0 1 t ( )  F 4sin − 2sin2 x t ⎯→

Chapter 4 Fourier transform 54.3.2 Time Shifting x()<">X(ja) x(t-t0)<"→>X(jio)eo Example 4.9 x 3/2 01234 t 17

17 Chapter 4 Fourier Transform §4.3.2 Time Shifting x(t)⎯→X(j) F ( ) ( ) F 0 0 j t x t t X j e   ⎯→ − − Example 4.9 0 1 2 3 4 t 3/ 2 1 x(t)

Chapter 4 Fourier transform 943.3 Conjugation and Conjugate Symmetry x(a)<"→>X(jo) x()=x()→X(o)=X( Rex(jo)=Rexljo) x()=x() x(ja)=-Imx(ja) X(jo=X(io) x()=x()→ ∠X(-jio)=-∠X(io)

18 Chapter 4 Fourier Transform §4.3.3 Conjugation and Conjugate Symmetry x (t)⎯→X (− j)  F  x(t) = x (t) X(j) = X (− j)   x(t) = x  (t) ImX(− j)= −ImX(j) ReX(− j)= ReX(j) x(t) = x  (t) X(− j) = X(j) X(− j)= −X(j)

Chapter 4 Fourier transform real even X Uja)real even x(t) real odd L X(o) Pure od inary ma Evkx(t)F>Rexjo) Odx()k F>jImxljo) Example em-=e"u()+e"u(-

19 Chapter 4 Fourier Transform x(t) real even X(j) real even x(t) real odd Purely imaginary odd X(j) Evx(t) ReX(j) ⎯F → Odx(t) jImX(j) ⎯F → Example e e u(t) e u( t) at at a t = + − − −

Chapter 4 Fourier transform 94.3.4 Differentiation and Integration 1. Differentiation (t) →jo(jo) dt Foxo d t 2. Integration Z F Ddre F>x(05(o)+.X( J0 Vo 20

20 ( ) (j) X(j) dt d x t F n n n ⎯→ Chapter 4 Fourier Transform §4.3.4 Differentiation and Integration 1. Differentiation ( ) jX(j) dt dx t ⎯F → 2. Integration ( ) ( ) ( ) ( )       X j j x d X F t 1 ⎯→ 0 + −